HSC Papers, Sorted by Topic (1984-1997), Part 1 of 4
Short Description
HSC Papers, Sorted by Topic (1984-1997), Part 1 of 4...
Description
4 UNIT MATHEMATICS – GRAPHS – HSC
Graphs 4U97-3b)! Let f(x) = 3x5 - 10x3 + 16x. i. Show that f (x) 1 for all x. ii. For what values of x is f (x) positive? iii. Sketch the graph of y = f(x), indicating any turning points and points of inflection.¤
y
Inflection points
9
-1 0
1
x
-9
»
« i) Proof ii) -1 < x < 0 or x > 1 iii) 4U96-4b)! i. ii.
1 3
On the same set of axes, sketch and label clearly the graphs of the functions y x and y ex . Hence, on a different set of axes, without using calculus, sketch and label clearly the graph of 1
the function y x 3 ex . 1
iii.
Use your sketch to determine for which values of m the equation x 3 ex mx 1 has exactly one solution.¤ y y = ex y
1 x3
y= 1 0
x
x (-1, -0.36)
« i)
ii)
4U95-3a)! Let f(x) = - x² + 6x - 8. On separate diagrams, and without using calculus, sketch the following graphs. Indicate clearly any asymptotes and intercepts with the axes. i. y = f(x) ii. y = │f(x)│ iii. y² = f(x) iv.
y=
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
1 f(x)
»
4 UNIT MATHEMATICS – GRAPHS – HSC
v.
f(x)
y=e .¤ y
y (3, 1) 2
4
x
y = f(x)
8
y = f(x)
(3, 1)
-8
2
« (i)
4
x
(ii) y y
y
f(x)
asymptote
(3, 1)
1 0
1
y2 = f(x) 2
3
4
18
2 y
x
4
1
x y
f(x)
1 f(x)
-1
(iv)
(iii)
y e
(3, e) y = ef(x) 3
(v) 4U94-5a)! Let f (x)
x asymptote
»
(x 2)(x 1) , for x 5 . 5 x 18 . 5 x
i.
Show that f (x) x 4
ii.
Explain why the graph of y = f(x) approaches that of y = -x - 4 as x approaches and as x approaches -. Find the values of x for which f(x) is positive, and the values of x for which f(x) is negative. Using part (i), show that the graph of y = f(x) has two stationary points. (There is no need to find the y coordinates of the stationary points.) Sketch the curve y = f(x). Label all asymptotes, and show the x intercepts. ¤
iii. iv. v.
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – GRAPHS – HSC
« (i) Proof (ii) As x approaches ,
18 approaches zero and so f(x) approaches -x - 4 (iii) Positive 5x
when x < -1, 2 < x < 5 and negative when -1 < x < 2, x > 5 (iv) Proof (v) y
-1
2
5
x
y = -x - 4 -4
asymptotes
» 4U93-4a)!
1 x x . On separate diagrams sketch the graphs of
Let f(x)
the following functions. For each graph label the asymptote. i. y = f(x) ii. y = f(│x│) iii. y = ef(x) iv. y² = f(x) Discuss the behaviour of the curve of (iv) at x = 1. ¤ y
y
1
-1
1
x
x
-1
-1 asymptotes
asymptotes
« (i)
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
(ii)
4 UNIT MATHEMATICS – GRAPHS – HSC
y
y
1 e
1
0 x
asymptotes
x
asymptote
(iv)
(iii)
NB: there is a vertical tangent at x = 1 »
4U92-4b)! Let f(x) = Ln(1 + x) - Ln(1 - x) where -1 < x < 1. i. Show that f (x) 0 for -1 < x < 1. ii. On the same diagram, sketch y = Ln(1 + x) for x > -1, y = Ln(1 - x) for x < 1 and y = f(x) for -1 < x < 1. Clearly label the three graphs. iii. Find an expression for the inverse function y f 1 ( x) . ¤ y y = f(x) y = ln(1 - x)
y = ln(1 + x) -1
1
0
x
(iii) y
y = f(x)
« (i) Proof (ii) 4U92-8a)!
x Consider the function f ( x) e 1 . 10 10
x
i. ii. iii.
Find the turning points of the graph of y f ( x) . Sketch the curve y = f(x) and label the turning points and any asymptotes. From your graph, deduce that
x e 1 10 x
iv.
10
for x < 10.
Using (iii), show that
11 10 e .¤ 10 9 10
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
10
ex 1 » ex 1
4 UNIT MATHEMATICS – GRAPHS – HSC
« (i) (0, 1) is a maximum turning point and (10, 0) is a minimum turning point. (ii) y
10
x
NB: The x-axis is an asymptote. (iii) Proof (iv) Proof » 4U91-4a)!
y 1 -4 -3 -2 -1 0 1 2 3 4 -1
x
The diagram is a sketch of the function y = f(x). On separate diagrams sketch: i. y = -f(x) ii. y = |f(x)| iii. y = f(|x|) iv. y = sin-1(f(x)). ¤ y
y
1
1 -3
-2
-1
1
2
x
3
-1
-3
-2
-1
1
2
3
x
(ii)
« (i)
(iii)
y
y
1
2
-3
-2
-1
1
2
3
x
-2
-1
2
2
-1
3
x
(iv)
»
4U91-4b)! The even function g is defined by
4e x 6e 2x g(x) g(x)
for x 0 for x 0
2 3
i.
Show that the graph of y = g(x) has a maximum turning point at Ln3, .
ii.
Sketch the curve y = g(x) and label the turning points, any points of inflexion, asymptotes, and points of intersection with the axes. Discuss the behaviour of the curve y = g(x) at x = 0. ¤
iii.
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – GRAPHS – HSC
y ( ln 3, ) 2 3
(ln 3, 23 )
( ln 6, 12 )
(ln 6, 12 ) ( ln 23 ,0)
(ln 23 ,0)
x
(0, -2) (iii) y = g(x) has a cusp at (0, -2) »
« (i) Proof (ii)
4U91-8a)!
for 0 x 2.
sinx Let f(x) x 1 i. ii.
for x 0
and prove that f ' is negative in this interval. 2 2x in this interval. ¤ Sketch the graph of y = f(x) for 0 x and deduce that sin x > 2 cos x(x tan x) « (i) f (x) (ii) x2 y
Find the derivative of f for 0 x
1
2
2
x
»
4U90-3b)! Consider the functions f, g defined by
x 1 , x 2 2 g(x) f(x) . f(x)
i. ii. iii. iv.
for x 2 ,
Sketch the hyperbola y = f(x), clearly labelling the horizontal and vertical asymptotes and the points of intersection with the x and y axes. Find all turning points of y = g(x). Using the same diagram as used in (i) sketch the curve y = g(x) clearly labelling it. On a separate diagram sketch the curve given by y = g(-x). ¤ y
y y = g(-x)
y = g(x)
y = g(x)
1 -2 y = f(x)
« (i) (iii) ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
1
1 4
-0.5
1 4
1
x
y = g(-x)
(ii) Min (1, 0) (iv)
2
x
»
4 UNIT MATHEMATICS – GRAPHS – HSC
4U90-4a)! f 4 (12, 2)
3
6
-4
10
12
t
(6, -4)
The diagram shows the graph of the function f, where
4 4 t , for 0 t 6 3 f(t) . t 10, for 6 t 12 The function F is defined for 0 x 12 by F(x)
x
f(t) dt . 0
i. Calculate F(6) and F(12). ii. Calculate those values of x for which F(x)=0. iii. Find all turning points of F. ¤ « (i) F(6) = 0, F(12) = -6 (ii) x = 0, 6 (iii) (3,6) is a relative maximum & (10, -8) is a relative minimum » 4U89-6a)!
y y = f '(x)
-3
-2
-1
0
1
2
3
x
The function f(x) has derivative f '(x) whose graph appears above. You are given that f (2) f (1) 0, f (x) approaches as x approaches and f (x) approaches 0 as x approaches . i. Sketch the graph of f(x) showing its behaviour at its stationary points. You are given that f(0) = 0 and f(3) > 0. ii. Describe the behaviour of f(x) as x approaches . ¤
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – GRAPHS – HSC
y MAX
-3
-2
-1
0
1
3 x
2
MIN
(ii) As x approaches , f(x) approaches an horizontal asymptote. As x approaches -, f(x) approaches -. »
« (i) 4U89-6b)!
4 3 2 Sketch the graph of g(x) x 4x 4x
i. ii.
1 showing that it has four real zeros. 2
On different diagrams sketch the curves: . y = |g(x)|; . y2 = g(x). . Indicate the nature of the curve y = |g(x)| at a zero of g(x). . Calculate the slope of the curve y2 = g(x) at any point x and describe the nature of the curve at a zero of g(x). ¤
iii.
y
y
y = g(x)
1
1
2
x
-1
1
2
3
x
2
« (i)
(ii) ()
()
y
y2 = g(x)
-1
1
2
3
x
(iii) () Since the curve is reflected in the x-axis, the curve has sharp points at zeros. () 4U88-2)! a.
dy g(x) . The curve has a vertical tangent. » dx 2 g(x)
Draw a neat sketch of the function f(x) = (x - 2)(6 - x). State the co-ordinates of its vertex and of its points of intersection with both co-ordinate axes.
16 . Clearly (x 2)(6 x)
b.
Hence or otherwise draw a neat sketch of the function g(x)
c.
indicate on your sketch the equations of the vertical asymptotes and the co-ordinates of any stationary points. The lines with equations x = 3 and x = 5 cut the graph of y = g(x) at P and Q respectively. Mark on your sketch the co-ordinates of P and Q. Shade the region R bounded by y = g(x) and the line PQ.
d.
Prove that the area of R is
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
32 8 log e 3 square units. ¤ 3
4 UNIT MATHEMATICS – GRAPHS – HSC
x=2
y
x=6 y = g(x)
16 3
P(3, 163 )
Q(5, 163 ) (4, 4) y = f(x)
y = g(x)
43
2
x
6 y = g(x)
« (a) (b) (c)
(d) Proof »
4U87-3) a.
A function f(x) is defined by f(x)
log e x for x > 0. x
Prove that the graph of f(x) has a relative maximum turning point at x = e and a point of 3
b. c. d.
inflexion at x = e 2 Discuss the behaviour of f(x) in the neighbourhood of x = 0 and for large values of x. Hence draw a clear sketch of f(x) indicating on it all these features. Draw separate sketches of the graphs of: . . (Hint:
e.
log e x ; x x . y log e x
y
There is no need to find any further derivatives to answer this part.)
What is the range of the function y
x ?¤ log e x
« (a) Proof (b) As x approaches 0, f(x) approaches -. As x approaches , f(x) approaches 0. (c) y
y 1 e, e
1
1 e, e
23 3 23 e , e 2
f ( x)
1
x
loge x x
23 3 23 e , e 2
f ( x)
(d) ()
log e x x
x
()
y
f (x)
x log e x
(e, e) x
(e) y < 0 and y e » 4U85-2ii) a. b.
Sketch the function g(x) = xe-x, for x -1. Given g(x) as in (a) above, the function f(x) is given by the rule
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – GRAPHS – HSC
g(x 2), f (x) g(x),
x 1 . x 1
Find the zeros of this function, and the maximum and minimum values. Draw a sketch of the graph of y = f(x). ¤ y
1, 1e
2, y = xe , x -1 2 e2
-x
-1, 1e
- 2, 2 e2
y
3, 1e
4, 2 e2
x
-1
0
1
x
2
(1, -e) (-1, -e)
(b)
« (a) 4U84-2i) Sketch the graphs of: a. (x + 3)(y - 2) = 1; b. x2 + y2 + 1 = 2(x + y). ¤
»
y
y
x2 + y2 + 1 = 2(x + y)
(x + 3)(y - 2) = 1 2
1 3
1
C(1, 1)
2 1 3
« (a)
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
1 2
-3
0
x
(b)
0
x
»
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
Complex Numbers 1: 1997 - 1991 4U97-2a)! i. ii.
Express
3 i in modulus-argument form.
Hence evaluate
6
3 i .¤ « i) 2cis
4U97-2b)! i. ii.
11 i ii) -64 » 6
Simplify (2i)3. Hence find all complex numbers z such that z3 = 8i. Express your answers in the form x + iy.¤ « i) 8i ii) 2i, 3 i, - 3 i »
4U97-2c)! Sketch the region where the inequalities z - 3 + i 5 and z +1 z -1 both hold.¤ Y 3 (3, -1)
X
-5
«
»
4U97-2d)! Let w = i. ii.
3 + 4i 5 + 12i and z = , so that w z 1. 5 13 Find wz and wz in the form x + iy. Hence find two distinct ways of writing 652 as the sum a2 + b2, where a and b are integers and 0 < a < b.¤ « i)
33 56 63 16 i, i ii) 652 332 562 162 632 » 65 65 65 65
4U97-8b)!
U
T
O
S R
The diagram shows points O, R, S, T and U in the complex plane. These points correspond to the complex numbers 0, r, s, t and u respectively. The triangles ORS and OTU are equilateral. Let
cos
i sin . 3 3
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
i. ii. iii.
Explain why u = t . Find the complex number r in terms of s. Using complex numbers, show that the lengths of RT and SU are equal.¤ « i) Rotation
s anticlockwise and UOT = ii) r = iii) Proof » 3 3
4U96-2a)! Suppose that c is a real number, and that z c i . Express the following in the form x iy , where x and y are real numbers: (iz) ; i. ii.
1 .¤ z « i) 1– ci ii)
c i » 2 1 c 1 c2
4U96-2b)! On an Argand diagram, shade the region specified by both the conditions Re(z) 4 and z 4 5i 3 .¤
y
Re(z) = 4 x |z – (4 – 5i)| = 3
O
4 – 5i
« 4U96-2c)! i. ii. iii.
»
Prove by induction that (cos i sin ) n cos(n) i sin(n) for all integers n 1 . Express w 3 i in modulus-argument form. Hence express w5 in the form x iy , where x and y are real numbers.¤ « i) Proof ii) 2 cis
4U96-2d)!
z
-1
O
3
The diagram shows the locus of points z in the complex plane such that ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
iii) –16 3 – 16i » 6
arg(z 3) arg(z 1)
. 3
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
This locus is part of a circle. The angle between the lines from -1 to z and from 3 to z is , as shown. Copy this diagram into your Writing Booklet.
. 3
i.
Explain why
ii.
Find the centre of the circle.¤
« i) The exterior angle of a triangle is equal to the sum of the interior opposite angles. ii) (1, 4U96-8a)! Let w = cos
2 3 )» 3
2 2 i sin . 9 9
i. ii.
Show that w k is a solution of z 9 - 1 = 0, where k is an integer. Prove that w + w2 + w3 + w4 + w5 + w6 + w7 + w8 = -1.
iii.
Hence show that cos cos
9
2 4 1 cos .¤ 9 9 8
« Proof » 4U95-2a)! Let w1 = 8 - 2i and w2 = - 5 + 3i. Find w1 w2 .¤ « 3 - 5i » 4U95-2b)! i. ii.
Show that (1 - 2i)² = - 3 - 4i Hence solve the equation z² - 5z + (7 + i) = 0.¤ « (i) Proof (ii) Z = 3 - i or 2 + i »
4U95-2c)! Sketch the locus of z satisfying:
3 ; 4
i.
arg(z - 4) =
ii.
Im z = │z│.¤ Im(z)
Im(z)
4 z 3 4
« (i) 4U95-2d)!
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4
Re(z)
(ii)
Re(z)
»
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
Q P O
The diagram shows a complex plane with origin 0. The points P and Q represent arbitrary non - zero complex numbers z and w respectively. Thus the length of PQ is │z - w│. i. Copy the diagram into your Writing Booklet, and use it to show that │z - w│ │z│ + │w│. ii. Construct the point R representing z + w. What can be said about the quadrilateral OPRQ? iii.
If │z - w│ = │z + w│, what can be said about the complex number
w ?¤ z
R Q P O OPRQ is a parallelogram (iii)
« (i) (ii) 4U95-4a)! i. ii.
w is imaginary » z
4 4 isin is a solution of zk = 1. 7 7
Find the least positive integer k such that cos
Show that if the complex number w is a solution of zn = 1, then so is wm, where m and n are arbitrary integers.¤ « (i) k = 7 (ii) Proof »
4U94-2a)! Let z = a + ib, where a and b are real. Find: i. Im(4i - z); ii. (3iz) in the form of x + iy, where x and y are real; tan , where arg(z 2 ) .¤ iii. « (i) 4 - b (ii) -3b - 3ai (iii) 4U94-2b)! Express in modulus - argument form: i. -1 + i; (1 i) n , where n is a positive integer.¤ ii. « (i) 4U94-2c)! i.
2ab » a b2 2
On the same diagram, draw a neat sketch of the locus specified by each of the following:
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
n 2 cos 3 i sin 3 (ii) 2 2 cos 3n4 i sin 3n4 » 4 4
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
. ii. iii.
z (3 2i) 2 z3 z5 .
. Hence write down all values of z which satisfy simultaneously z (3 2i) 2 and z 3 z 5 . Use your diagram in (i) to determine the values of k for which the simultaneous equations z (3 2i) 2 and z 2i k have exactly one solution for z.¤ Im(z) 4
(3 + 2i)
« (i)
1
5 Re(z)
(ii) z = 1 + 2i (iii) k = 1, 5 »
4U94-4a)! Find and , given that z3 3z 2i (z ) 2 (z ) .¤ « = -i, = 2i » 4U94-7a)! i.
It is known that if f '(x) 0 and f(0) = 0, then f (x) 0 for x > 0. Show that
3 2 sin x x x 0 for x > 0, and hence show that sin x 1 x for x > 0. 6 x 6 Let the points A0 , A1 , A2 ,...., An 1 represent the nth roots of unity on an Argand diagram, where A k 2k 2k represents cos . Let P be the regular polygon A 0 A1 ... A n1 . i sin n n n 2 ii. Show that the area of P is sin . 2 n iii. Using part (i), or otherwise, show that for all n 26 , P covers more than 99% of the unit
circle.¤ « Proof » 4U94-8b)! Let x = be a root of the quartic polynomial P(x) x4 Ax3 Bx 2 Ax 1 , where A and B are real. Note that may be complex. i. Show that 0.
1 A x 1 B . x x2
ii.
Show that x = is also a root of Q( x) x 2
iii.
With u x , show that Q(x) becomes R(u) u2 Au (B 2) .
iv.
For certain values of A and B, P(x) has no real roots. Let D be the region of the AB plane where P(x) has no real roots for A 0.
1 x
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
B
c
T
L
0
A
-2
The region D is shaded in the figure. Specify the bounding straight-line segment L and curved segment c. Determine the coordinates of T.¤ « (i) Proof (ii) Proof (iii) Proof (iv) The straight line segment is B = 2A - 2 and the curved segment is B 14 A2 2 . T is the point (4, 6). » 4U93-2a)! i. On an Argand diagram, shade in the region determined by the inequalities 2 Im z 4 and 6 arg z 4 . ii.
Let z0 be the complex number of maximum modulus satisfying the inequalities of (i). Express z0 in the form a + ib.¤
y 4 2 6
4
x
« (i)
(ii) z 0 4 3 4i »
4U93-2b)! Let be a real number and consider (cos + i sin )3. i. Prove cos 3 cos3 3cos sin2 . ii. Find a similar expression for sin 3.¤ « (i) Proof (ii) sin 3 3sin cos2 sin 3 » 4U93-2c)!
z 4 0 .¤ z
Find the equation, in Cartesian form, of the locus of the point z if Re
« x2 + y2 - 4x = 0, excluding the origin (0, 0) » 4U93-2d)! By substituting appropriate values of z1 and z2 into the equation arg
1 tan 1 2 tan 1 .¤ 4 3 ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
z1 arg z1 arg z2 show that z2
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
« Proof » 4U93-2e)! Let P, Q, and R represent the complex numbers w1, w2 and w3 respectively. What geometric properties characterise triangle PQR if w2 - w1 = i(w3 - w1)? Give reasons for your answer.¤ « PQR is a right-angled isosceles triangle, with RPQ
and PQ=PR» 2
4U93-8a)! Let the points A1, A2, ..., An represent the nth rots of unity, w1, w2, ..., wn, and suppose P represents any complex number z such that │z│ = 1. i. Prove that w1 + w2 + ... + wn = 0. ii. Show that PA i 2 (z wi )(z wi ) for i = 1, 2, ..., n. n
iii.
Prove that
PA z 1
2 i
2n .¤ « Proof »
4U92-2a)! The points A and B represent the complex numbers 3 - 2i and 1 + i respectively. i. Plot the points A and B on an Argand diagram and mark the point P such that OAPB is a parallelogram. ii. What complex number does P represent?¤ y 2 1
B
0
1
2
3
-1 -2
« (i)
4
x
P(4, -1) A
(ii) 4 - i »
4U92-2b)! Let z = a + ib where a 2 b2 0. i
1 Show that if Im(z) > 0 then Im 0 .
ii.
Prove that
z
1 1 .¤ z z « Proof »
4U92-2c)! Describe and sketch the locus of those points z such that: zi zi i. z i 2 z i .¤ ii.
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
y i
x -i
« (i) The locus is the real axis.
(ii) The locus is a circle, centre (0, -3) and y
x -3
» radius 2 2 . 4U92-2d)! It is given that 1 + i is a root of P(z) 2z3 3z2 rz s where r and s are real numbers. i. Explain why 1 - i is also a root of P(z). ii. Factorise P(z) over the real numbers.¤ « (i) If z1 is a root of P(z) = 0 then z1 is also a root. Thus, if (1 + i) is a root then 1 i 1 i is also a root. (ii) P(z) (2z 1)(z2 2z 2) » 4U92-7b)! Suppose that z7 1 where z 1. i. ii. iii.
1 1 1 0. z z2 z3
Deduce that z3 z2 z 1
1 reduce the equation in (i) to a cubic equation in x. z 2 3 1 Hence deduce that cos cos cos .¤ 7 7 7 8 By letting x z
« (i) Proof (ii) x3 + x2 - 2x - 1 = 0 (iii) Proof » 4U91-2a)! Plot on an Argand diagram the points P, Q, and R which correspond to the complex numbers 2i, i, and - 3 - i, respectively. Prove that P, Q, and R are the vertices of an equilateral triangle.¤
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
3-
4 UNIT MATHEMATICS – COMPLEX NUMBERS 1 – HSC
y 3 2
P(0, 2)
1
-2
«
-1
R( 3, 1)
1 -1
2
3
Q( 3, 1)
x
»
4U91-2b)! Let z1 = cos1 + isin1 and z2 = cos2 + isin2, where 1 and 2 are real. Show that: i.
1 = cos1 - isin1 z1
ii.
z1z2 = cos(1 + 2) + isin(1 + 2).¤ « Proof »
4U91-2c)! i. ii.
Find all pairs of integers x and y such that (x + iy)2 = -3 - 4i. Using (i), or otherwise, solve the quadratic equation z 2 3z (3 i) 0 .¤ « (i) x = 1, y = -2 and x = -1, y = 2 (ii) z = 2 - i or 1 i »
4U91-2d)!
C D B
F A
E
0 In the Argand diagram, ABCD is a square, and OE and OF are parallel and equal in length to AB and AD respectively. The vertices A and B correspond to the complex numbers w1 and w2 respectively. i. Explain why the point E corresponds to w2 - w1. ii. What complex number corresponds to the point F? iii. What complex number corresponds to the vertex D?¤ « (i) Proof (ii) i(w2 - w1) (iii) w1(1 - i) + iw2 »
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
Complex Numbers 2: 1990 - 1984 4U90-1a)! Let z = a+ib, where a and b are real numbers and a 0 . i. Express |z| and tan(arg z) in terms of a and b. ii.
Express
z in the form x+iy, where x and y are real. ¤ 3 5i « (i) z
a 2 b 2 , tan(arg z)
b (3a 5b) i(5a 3b) (ii) » a 34 34
4U90-1b)! i. ii.
1 i 3 3 show that w 1 . 2 10 Hence calculate w . ¤
If w
« (i) Proof (ii)
1 i 3 » 2 2
4U90-1c)!
1 in modulus argument form. ¤ z « z 5 2 cos i sin , z 2 50cos i sin , 4 2 4 2 1 1 cos isin » 4 z 5 2 4
If z = 5 - 5i write z, z2 and
4U90-1d)! Let u and v be two complex numbers, where u = -2+i, and v is defined by |v| = 3 and argv i. ii.
. 3
On an Argand diagram plot the points A and B representing the complex numbers u and v respectively. Plot the points C and D represented by the complex numbers u-v and iu, respectively. Indicate any geometric relationships between the four points A, B, C, and D. ¤
y 3
B 3
A O C «
x
D »
4U89-1a)! Evaluate |2 + 3i|. ¤ « 13 »
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
4U89-1b)! Given that a and b are real numbers, express in the form x + iy, where x and y are real: (a bi)(5 i) ; i.
a bi .¤ 3 4i
ii.
« (i) (5a + b) + (5b - a)i (ii)
3a 4b 3b 4a i» 25 25
4U89-1c)! Find the complex square roots of 10 - 24i, giving your answers in the form x + iy, where x and y are real. ¤ « (3 2 2 2i) » 4U89-1d)! On and Argand diagram shade in the region containing all points representing complex numbers z such that 2 Rez 4 and 1 Imz 3 . ¤ y 3
2
4
x
-1
»
« 4U89-1e)! Find in modulus-argument form all complex numbers z such that z3 = -1 and plot them on an Argand diagram. ¤ y z1 3
x
z2
z3
3
3
« z1 cos isin , z2 cos isin , z3 cos
5 isin 5 ; 3 3
4U89-1f)! On separate diagrams draw a neat sketch of the locus specified by: i. ii.
arg z(z (1 3 i)) ; 3 2 2 z z 16i . ¤
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
»
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
y
y
z xy = 4
3
x
3
1
x
(ii)
« (i) 4U88-4a)! i. ii.
Express z = 2 i 2 in modulus-argument form. Hence write z22 in the form a + ib, where a and b are real. ¤
« (i) z 2 cos i sin (ii) 222i »
4U88-4b)! i.
4
4
On an Argand diagram shade in the region R containing all points representing complex numbers z such that 1 < |z| < 2 and
ii.
»
argz . 4 2
In R mark with a dot the point K representing a complex number z. Clearly indicate on your diagram the points M, N, P and Q representing the complex numbers z , -z,
1 and 2z z
respectively. ¤ y
Q
2 K
1
1
4
2 x
P N
M
»
« 4U88-4c)! Show that the locus specified by 3|z - (4 + 4i)| = |z - (12 + 12i)| is a circle. Write down its radius and the co-ordinates of its centre. Draw a neat sketch of the circle. ¤ Im(z) 6
(3, 3)
« Centre is (3, 3) and radius is 3 2 . ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
O
6
R(z)
»
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
4U87-4i) Find the complex square roots of 7 6 2i giving your answers in the form x + iy, where x and y are real. ¤ « 3 i 2 and (3 i 2) » 4U87-4ii) Let z1 = 4 + 8i and z2 = -4 - 8i. a. Draw a neat sketch of the locus specified by |z - z1| = |z - z2|. b. Show that the locus specified by |z - z1| = 3|z - z2| is a circle. Give its centre and radius. ¤ Im(z) z1
P(z)
Re(z)
z2
« (a) (b) Proof, centre is (-5 - 10i) and radius is 3 5 units. » 4U87-4iii) a. Let OABC be a square on an Argand diagram where O is the origin. The points A and C represent the complex numbers z and iz respectively. Find the complex number representing B. b. The square is now rotated about O through 45° in an anticlockwise direction to OABC. Find the complex numbers representing A, B and C. ¤ « (a) (1 + i)z (b)
1 1 (1 i)z , i 2z and (-1 i)z represent A, B and C respectively. » 2 2
4U86-4i) Given that z1 = 3 - i, z2 = 2 + 5i, express in the form a + ib, where a, b are real, (z1 )2 ; a. b.
z1 ; z2
c.
z1 .¤ z2 « (a) 8 + 6i (b)
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
1 17 i (c) 29 29
10 » 29
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
4U86-4ii) Given that for the complex number z, |z| = 2, arg z = a. b.
2 write in the form a + ib, where a, b are real: 5
z; z7. ¤ « (a) z 2 cos
4U86-4iii) a.
Draw a sketch of the region of the Argand diagram consisting of the set of all values of z for which 1 |z| 4 and
b.
2 2 (b) z7 27 (-cos i sin ) » i 2sin 5 5 5 5
. .
3 arg z . 4 4
The curve in the Argand diagram for which |z - 2| + |z - 4| = 10 is an ellipse. Write down the coordinates of the centre, and the lengths of the major and minor axes of this ellipse. On a separate Argand diagram, show the region for which z satisfies the inequalities z + z 6 or |z - 2| + |z - 4| 10. ¤ y
4
1
« (a)
4
x
(b) () Centre is (3, 0). Major axis is 10 units and Minor axis y
-2
is 4 6 units. () 4U85-3i) Reduce the complex expression
8
x
»
(2 i)(8 3i) to the form a + ib, where a, b are real numbers. ¤ (3 i) 11 5 « i» 2 2
4U85-3ii) The complex number z is given by z 3 i . a. Write down the values of arg z and |z|. b. Hence, or otherwise, show that z7 + 64z = 0. ¤ « (a) arg z 4U85-3iii) On the Argand diagram, let A = 3 + 4i, B = 9 + 4i. ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
3
5 , z = 2 (b) Proof » 6
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
a.
Draw a clear sketch to show the important features of the curve defined by |z - A| = 5. Also, for z on this curve, find the maximum value of |z|. On a separate diagram, draw a clear sketch to show the important features of the curve defined by |z - A| + |z - B| = 12. For z on this curve, find the greatest value of arg z. ¤
b.
y
A(3 + 4i) 5
4
3 z - a = 5
x
Maximum value of z = 10. (b)
« (a) y
B(9 + 4i)
A(3 + 4i) 6 + 4i
x
z - A + z - B = 12
Greatest value of arg z
.» 2
4U84-3i) Calculate the modulus and argument of the product of the roots of the equation (5 + 3i)z2 - (1 - 4i)z + (8 - 2i) = 0. ¤ « P
2 , arg P
» 4
4U84-3ii) Let A = 1 + i, B = 2 - i. Draw sketches to show the loci specified on the Argand diagram by a.
arg(z - A) =
b.
4
|z - A| = |z - B|. ¤ y
y P(z)
A(1 + i)
P(z) Locus of z 1
0
A(1 + i)
x
4
0
« (a) 4U84-3iii)
1 arg(z - A)
Show that the point representing cos
x
(b)
Locus of z
B(2 - i)
»
i sin on the Argand diagram lies on the circle of radius one 3 3
with centre at the point which represents 1. ¤ « Proof » 4U84-3iv) ¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – COMPLEX NUMBERS 2 – HSC
R is a positive real number and z1, z2 are complex numbers. Show that the points on the Argand diagram which represent respectively the numbers z1, z2, angled triangle. ¤
z1 iRz 2 , form the vertices of a right 1 iR
« Proof »
¤©BOARD OF STUDIES NSW 1984 - 1997 ©EDUDATA: DATAVER1.0 1995
4 UNIT MATHEMATICS – CONICS – CSSA
Conics 4U96-4)!
x2 y2 The diameter of the ellipse 2 2 1 (where a > b > 0) through the point P(a cos , a b
a.
b sin ) meets the circle x2 + y2 = a2 at the point R(a cos , a sin ). i. Show this information on a sketch.
b tan . a
ii.
Show that tan =
iii. iv. v.
Prove that the tangent to the ellipse at P has equation bx + ay tan = ab sec . Show that the tangent to the circle at R has equation ax + by tan = a2sec . If the tangent to the ellipse at P and the tangent to the circle at R are concurrent with the right hand directrix of the ellipse, show that sec =
2 , where e is the e
eccentricity of the ellipse.†
x2 y2 The diameter of the ellipse 1 through the point P on the ellipse meets the circle 25 9
b.
x2 + y2 = 25 at R. Tangents to the ellipse at P and the circle at R are concurrent with a directrix of the ellipse. Using the results from part (a): If P lies in the first quadrant, i. find the coordinates of P and R. ii. Sketch the ellipse and the circle, showing the coordinates of P and R, and the point of intersection of the tangents and the appropriate directrix.† a R b P -a
a
x
-b
« a) i)
-a
ii) iii) iv) v) Proof
3 21 50 15 21 , R , ii) 5 17 17
b) i) P 2,
Y
5
25 5 21 , 4 14
R 3 P -5
x
5 -3
x
25 4
-5
4U95-4)! The hyperbola xy = c2 meets the ellipse
c c x2 y2 2 1 at P ct1, and Q ct 2 , where t1 > t2 > 0. 2 t2 t1 a b
Tangents to the hyperbola at P and Q meet in T, while tangents to the ellipse at P and Q meet in V. i. Show this information on a sketch. ii.
c t
Show that the parameter t of a point ct , where xy = c2 and satisfies the equation b2c2t4 - a2b2t2 + a2c2=0.
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
x2 y2 1 intersect a2 b2
»
4 UNIT MATHEMATICS – CONICS – CSSA
iii.
c t
Using without proof the result that the tangent to hyperbola xy = c2 at the point ct , has
2ct1t 2 2c , . t1 t 2 t1 t 2
equation x + t2y = 2ct, show that T has coordinates
x2 y2 1 at the point (x1, y1) has a2 b2 a2 b2 t 1t 2 2 2 2 2 , equation b x1x + a y1y = a b , , show that V has co-ordinates c( t 1 t 2 ) ( t 1 t 2 )
iv.
Using without proof the result that the tangent to
v. vi.
Show that TV passes through the origin. Show that if V lies at a focus of the hyperbola, then the ellipse is in fact a circle and find the radius of this circle in terms of c.†
y
xy = c2 Q
V P
T
x x2 y2 1 a 2 b2
« i)
ii) iii) iv) v) Proof vi) Proof Radius = c 1 5 »
4U94-4)! P(20cos , 12sin ) is a point on the ellipse
x2 y2 1. 202 122
P lies in the first quadrant, and the tangent to the ellipse at P meets the directrices in Q and R where Q is nearer the focus S and R is nearer the focus S. Q and R each lie above the x axis, and QS meets RS in K where K lies in the third quadrant. i. Sketch the ellipse showing it’s directrices and foci and the points P, Q, R and K. ii. Show that the tangent at P has equation 3x cos + 5y sin = 60.
4(9 25sin2 ) . 3sin
iii.
Show that K has co-ordinates 20 cos ,
iv.
If K lies on the ellipse, find the co-ordinates of P and show that PSKS is a rectangle.† y Q P(20cos, 12sin) 12 R -20 x = -25
« i) 4U93-4a)!
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
-16
O
-12
16
20
x x = 25
ii) iii) Proof iv) k(-5 7 , -9), Proof »
4 UNIT MATHEMATICS – CONICS – CSSA 2
x y 2 2 1 a b
y
-a
2
0 a
ae
x (x - ae)2 + y2 = a2(e2 + 1)
i.
Show that the tangent at P(a sec , b tan ) on the hyperbola
x sec y tan 1 0 . a b ii.
x2 y2 1 has equation a2 b2
Show that if the tangent at P is also a tangent to the circle with centre (ae, 0) and radius
a e2 1 , then sec = -e. iii.
iv.
Deduce that the points of contact P, Q on the hyperbola of the common tangents to the circle and hyperbola are the extremities of a latus rectum of the hyperbola, and state the coordinates of P and Q. Find the equations of the common tangents to the circle and hyperbola, and find the coordinates of their points of contact with the circle.† « i) ii) Proof iii) P(-ae, b e2 1 ), Q(-ae, -b e2 1 ) iv) xe y + a = 0, (0, a) »
4U92-4b)! Consider the ellipse i. ii.
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
x2 y2 1 , where a > b > 0. a2 b2
Show that the tangent to the ellipse at the point P(a cos , b sin ) has equation bx cos + ay sin - ab = 0. R and R are the feet of the perpendiculars from the foci S and S on to the tangent at P. Show that SR.SR = b2.† « Proof »
4 UNIT MATHEMATICS – CONICS – CSSA
4U91-4)! a.
x2 y2 P(a cos , b sin ), Q(a sec , b tan ) lie on the ellipse 2 2 1 and the hyperbola a b 2 2 x y 1 respectively. M and N are the feet of the perpendiculars from P, Q respectively a2 b2 to the x axis. O < < , and QP produced meets the x axis in K. A is the point with 2 coordinates (a, 0).
KM cos , KN
i.
Using without proof the similarity of KPM and KQN, show that
ii.
and hence show that K has coordinates (-a, 0). Sketch the ellipse and hyperbola showing the positions of P, Q, M, N, A and K.
iii.
Show that the tangent to the ellipse at P has equation
x cos y sin 1 and a b
deduce that this tangent passes through N.
x sec y tan 1, a b
iv.
Given that the tangent to the hyperbola at Q has equation
v.
show that this tangent passes through M. Show that the tangents PN and QM and the common tangent at A are concurrent,
2
and show that the point of concurrence is T(a, b tan ). vi. b.
If the common tangent at A meets QP in V, show that T is the midpoint of AV.†
The result in (a) (i) provides a method of constructing the hyperbola
x2 y2 1 from the 4
x2 auxiliary circle x + y = 4, and the ellipse y2 1 . Indicate why this is so on a new 4 2
2
sketch by using the auxiliary circle to construct one such pair of points P, Q each with parameter , 0 < <
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
.† 2
4 UNIT MATHEMATICS – CONICS – CSSA
y b
Q
P
x -a
K
M
a
A
N
-b
« a) i) Proof ii)
iii) iv) v) vi) y 2 Q 1
P
-2
M
2
N
x
-1
Proof b)
-2
»
4U90-4)! a.
Show that the tangent to the ellipse
x2 y2 1 at the point P(x1, y1) has cartesian equation 9 4
xx1 yy1 1. 9 4 b.
Show that if tangents are drawn from a point W(x0, y0) external to the ellipse
x2 y2 1, 9 4
touching the ellipse at P, Q respectively, then the equation of the chord of contact PQ is
xx0 yy0 1. 9 4
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
4 UNIT MATHEMATICS – CONICS – CSSA
c. y
2
y=2
2 x2 y 1 9 4
-3
P(x1, y1)
0
3
(x - 7)2 + y2 = 4 5
9
7
x
R(x2, y2) y = -2
-2 y = mx + k 2
The above diagram shows the ellipse
2
x y 1 and the circle (x - 7)2 + y2= 4. Clearly 9 4
y = 2 and y = -2 are common tangents to the ellipse and the circle. Suppose the line y = mx + k, m 0, is also a common tangent, touching the ellipse at P(x1, y1) and the circle at R(x2, y2) as shown. i. Copy the diagram and use symmetry to draw a fourth common tangent, touching the ellipse at Q and the circle at T, and write down the coordinates of Q and T, on your diagram. Deduce that the equation of QT is y = -mx - k.
k , 0 . m
ii.
PR and QT intersect at V. Show V has coordinates
iii.
Use the fact that PQ is the chord of contact of tangents from V to the ellipse to show that x1 =
9m . k
iv.
9m x2 (mx k)2 Deduce that x1 = is a double root of the equation 1 , and 9 4 k
v.
hence show that 9m2 - k2 + 4 = 0. Show that if y = mx + k is a tangent to the circle (x - 7)2 + y2 = 4, then 45m2 + k2 + 14mk - 4 = 0.
vi.
Show that
m 7 , and find the coordinates of P, Q and V, and the equations of k 27
the two oblique common tangents.†
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
4 UNIT MATHEMATICS – CONICS – CSSA
y 2
y=2 T(x2, -y2)
P(x1, y1)
-3
0
3
5
Q(x1, y1)
9
7 R(x2, -y2) y = -2
-2
ii) iii) iv)
« a) b) Proof c) i)
7 8 2 7 8 2 7x 2 2 y 27 7x 2 2y , Q , , V , 0 , 1, 1 » 7 27 9 27 9 3 9 3 9
v) Proof vi) P ,
4U90-5b)! A parabola passes through the points (-a, 0), (0, h) and (a, 0) where a > 0, h > 0. Show that the area enclosed by this parabola and the x axis is
4 ah.† 3 « Proof »
4U89-4a)! i. ii.
Show that the ellipse 4x2 + 9y2 = 36 and the hyperbola 4x2 - y2 = 4 intersect at right angles. Find the equation of the circle through the points of intersection of the two conics.† « i) Proof ii) x2 + y2 = 5 »
4U89-4b)! i.
Show that the tangent to the hyperbola
x2 y2 1 (where a > b > 0) at the point P(a sec, a 2 b2
b tan) has equation bx sec - ay tan = ab. ii.
If this tangent passes through a focus of the ellipse
x2 y2 1 (where a > b > 0) show that a2 b2
it is parallel to one of the lines y = x, y = -x and that its point of contact with the hyperbola lies on a directrix of the ellipse.† « Proof » 4U88-4)! P(2Ap, Ap2) is a point on the parabola x2 = 4Ay. Q(a cos, b sin) is a point on the ellipse
x2 y2 1. a2 b2 In what follows you may use without proof the results that the tangent to x2 = 4Ay at P and the tangent to
x cos y sin x2 y2 1 respectively. 2 1 at Q have equations px - y = Ap2 and 2 a b a b
a.
Using the fact that two lines are coincident if the corresponding coefficients are in proportion
x2 y2 deduce that the tangent to x = 4Ay at P is also the tangent to 2 2 1 at Q if a b b a cos = and sin = . Ap2 Ap 2
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
4 UNIT MATHEMATICS – CONICS – CSSA 2
b. c.
d.
Hence show that PQ is a common tangent to x2 = 4Ay and
x y 1 if a2 b2
A2p4 - a2p2 - b2 = 0, and deduce that there are exactly two such common tangents. Let p0 > 0 be the parameter of the point of contact of one of these common tangents with the parabola and let A > 0. Sketch the parabola, the ellipse and both common tangents showing, in terms of p0 the coordinates of the points of contact of the tangents with both curves and the intercepts of the tangents on the coordinate axes. Using symmetry sketch on the same diagram the parabola x2 = -4Ay and the two common tangents to x2 = -4Ay and
e.
2
x2 y2 1. a2 b2
What is the nature of the quadrilateral formed by the four tangents on this diagram? Deduce that this quadrilateral is a square if A2 = a2 + b2. Find the equation of the circle with centre (0, 0) for which the quadrilateral formed by the four tangents common to the circle and the curve x2 = 8y is a square.† y
x2 y2 1 a 2 b2
(2Ap0, Ap02)
(-2Ap0, Ap02) -Ap0
Ap0 O
a 2 b2 , Ap 0 Ap 0 2
x a 2 b2 , Ap0 Ap02
-Ap02
« a) Proof b) Proof c) d) 4U87-4i)!
p0x - y = Ap02
-p0x - y = Ap02
e) x2 + y2 = 2 »
a.
x2 y2 Show that the normal to the ellipse 2 2 1 (a2 > b2) at the point P(x1, y1) has equation a b
b.
a2y1x - b2x1y = (a2 - b2)x1y1. This normal meets the major axis of the ellipse at G. S is one focus of the ellipse. Show that GS = e.PS where e is the eccentricity of the ellipse.† « Proof »
4U87-4ii)! a.
By using the result in part (i)(a) above, or otherwise, show that the normal to the ellipse
x2 y2 1 at the point P(5 cos, 3 sin) has equation 5xsin - 3ysin = 16sin cos. 25 9 b.
c. †©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
This normal cuts the major and minor axes of the ellipse at G and H respectively. Show that as P moves on the ellipse the mid point of GH describes another ellipse with the same eccentricity as the first. On the same axes sketch the two ellipses showing clearly the coordinates of the intercepts on the coordinate axes.†
4 UNIT MATHEMATICS – CONICS – CSSA
y 3 2.6 -1.6
1.6
-5
5
x
-2.6 -3
« a) Proof b) Proof c) 4U86-4i)! Show that the curves x2 - y2 = c2 and xy = c2 cross at right angles.†
»
« Proof » 4U86-4ii)! Show that the tangent to the hyperbola
x2 y2 1 at the point P(a sec, b tan) has equation a2 b2
bx sec - ay tan = ab, and deduce that the normal there has equation by sec + ax tan = (a2 + b2) sectan. The tangent and the normal cut the y-axis at A and B respectively. Show that the circle on AB as diameter passes through the foci of the hyperbola. (It is enough to show that the circle passes through one focus and then to use symmetry).† « Proof » 4U85-4)! i.
Show that the point P (a sec , b tan ) lies on the hyperbola If Q is the point (a sec , b tan ) where + = is
ii.
x2 y2 1 for all values of . a2 b2
show that the locus of the midpoint of PQ 2
x2 y2 y .† a2 b2 b
Show that the equation of the normal to the hyperbola
x2 y2 1 at the point (a sec , b tan a2 b2
) is ax tan + by sec = (a2 + b2) sec tan . The ordinate at P meets an asymptote of the hyperbola at Q. The normal at P meets the x axis at G. Show that GQ is at right angles to the asymptote.† « Proof » 4U84-4i)!
x2 y2 1 is a 2 b2 x2 y2 c2 = a2m2 + b2. Show that the pair of tangents drawn from the point (3, 4) to the ellipse 1 16 9
Show that the condition for the line y = mx + c to be tangent to the ellipse
are at right angles to one another.† « Proof » 4U84-4ii)!
x2 y2 1 is a 2 b2 x2 y2 2 2 ax sin + by = (a + b )tan. The normal at the point P(a sec, b tan) on the hyperbola 2 2 1 a b Show that the equation of the normal at the point P(a sec, b tan) on the hyperbola
meets the x axis at G and PN is the perpendicular from P to the x axis. Prove that OG = e2.ON (where O is the origin).† †©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
4 UNIT MATHEMATICS – CONICS – CSSA
« Proof »
†©CSSA OF NSW 1984 - 1996 ©EDUDATA: DATAVER1.0 1996
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