Hyperbola Problems
Short Description
problems...
Description
26
Hyperbola
LEVEL–I 1.
Find the equation of the hyperbola with vertices at ( 5, 0) and foci at ( 7, 0).
2.
Find the equation of the hyperbola satisfying in the given conditions. Foci (0, 10 ) , passing through (2, 3).
3.
Find the centre, foci, directrices, length of the latus rectum, length & equations of the axes and the asymptotes of the hyperbola 10x 2 – 9y2 + 32x + 36y – 164 = 0.
4.
The hyperbola x 2/a2 – y2/b 2 = 1 passes through the point of intersection of the lines, 7x + 13y – 87 = 0 & 5x – 8y + 7 = 0 & the latus rectum is 32 2 / 5 . Find ‘a’ & ‘b’.
5.
In a rectangular hyperbola x2 – y2 = a2, prove that SP. SP CP 2 , where S and S are foci, C is the centre and P is any point on the hyperbola.
6.
The centre of a variable rectangular hyperbola lie on a line x + y = 3. A variable circle intersects a hyperbola in such a way that the mean value of points of intersection is always (3, 5). Find the locus of the centre of the variable circle.
7.
If the normals at four points P (xi, yi), i = 1, 2, 3, 4 on the rectangular hyperbola xy = c2, meet at the point Q(h, k), prove that (i) x1 + x2 + x3 + x4 = h (ii) y1 + y2 + y3 + y4 = k
8.
The tangent at P on the hyperbola
9.
A variable chord of the hyperbola
x 2 y2 1 meets one of the asymptote in Q. Show that the locus a 2 b2 of the mid point of PQ is a similar hyperbola. x 2 y2 1 is tangent to the circle x2 + y2 = c2. Prove that locus of a 2 b2
2
x 2 y2 x 2 y2 its mid point is 2 2 c 2 4 4 . b b a a
10.
If the tangent at the point (h, k) to the hyperbola (x 2/a2) – (y2/b2) = 1, cuts the auxiliary circle x2 + y2 = a2 at points whose ordinates y1 and y2 show that
1 2 1 + = . y1 y 2 k
Hyperbola
27
LEVEL–II 1.
If two points P & Q on the hyperbola x 2/a2 – y2/b2 = 1 whose centre is C be such that CP is perpendicular to CQ & a < b, then prove that
1 1 1 1 2 2. 2 2 CP CQ a b
2.
A rectangular hyperbola whose centre is C is cut by any circle of radius r at the four points P, Q, R, S. Prove that CP2 + CQ2 + CR2 + CS2 = 4r2.
3.
A tangent to the parabola x2 = 4 ay meets the hyperbola xy = k2 in two points P and Q. Prove that the middle point of PQ lies on a parabola.
4.
5.
x2 y2 The perpendicular from the centre upon the normal on any point of the hyperbola 2 2 1 a b meets at R. Find the locus of R.
b g
Tangents are drawn from the point , to the hyperbola 3x2 - 2y2 = 6 and are inclined at angles
and to the x - axis. If tan . tan 2 , prove that 2 2 2 7 . 6.
Tangents are drawn from any point on the rectangular hyperbola x 2 – y2 = a2 – b2 to the ellipse x2/a2 + y2/b2 = 1. Prove that these tangents are equally inclined to the asymptotes of the hyperbola.
7.
If a chord joining the points P a sec , a tan and Q a sec , a tan on the hyperbola
b
g
b
g
x2 - y2 = a2 is a normal to it at P, show that tan tan (4 sec 2 1) . 8.
Show that the locus of the middle points of normal chords of the rectangular hyperbola x2 – y2 = a2 is (y2 – x2)3 = 4a2 x2y2.
9.
Chords of the hyperbola x2/a2 – y2/b2 =1 are tangents to the circle drawn on the line joining the foci as diameter. Find the locus of the point of intersection of tangents at the extremities of the chords.
10.
A parallelogram is constructed with its sides parallel to the asymptotes of the hyperbola x2/a2 – y2/b2 = 1, and one of its diagonals is a chord of the hyperbola; show that the other diagonal passes through the centre.
28
Hyperbola
IIT JEE PROBLEMS
(OBJECTIVE)
A. 1.
Fill in the blanks An ellipse has eccentricity 1/2 and one focus at the point P(1/2, 1). Its one directrix is the common tangent, nearer to the point P, to the circle x 2 + y2 = 1 and the hyperbola x2 - y2 = 1. The equation of the ellipse in the standard form is................ [IIT - 96]
B. 1.
Multiple choice questions with one or more than correct answer. If the circle x2 + y2 = a2 intersects the hyperbola xy = c2 in four points P(x1, y1), Q(x2, y2), R(x3, y3), S(x4, y4), then (A) x1 + x2 + x3 + x4 = 0 (B) y1 + y2 + y3 + y4 = 0 4 (C) x1 x2 x3 x4 = c (D) y1 y2 y3 y4 = c4 [IIT - 98]
C.
Multiple choice questions with one correct answer.
1.
Identify the types of curves with represented by the equation (A) an ellipse
(B) a hyperbola
(C) a circle
x2 y2 1 , where r > 1 is 1 r 1 r (D) none of these [IIT - 81]
2.
The curve described parametrically by, x = t2 + t + 1, y = t2 - t + 1 represents : (A) a pair of straight lines (B) an ellipse (C) a parabola (D) a hyperbola
3.
Let P ( a sec , b tan ) and Q( a sec , b tan ) , where
[IIT - 99]
, be two points on the 2
x2 y2 1 . If (h, k) is the point of intersection of the normals at P and Q, then a 2 b2 k is equal to [IIT - 99] hyperbola
a 2 b2 (A) a
(B)
FG a H
2
b2 a
IJ K
(C)
a 2 b2 b
(D)
FG a H
2
b2 b
IJ K
4.
If x = 9 is the chord of contact of the hyperbola x 2 - y2 = 9, then the equation of the corre sponding pair of tangents, is : [IIT - 99] 2 2 2 2 (A) 9x - 8y + 18x - 9 = 0 (B) 9x - 8y - 18x + 9 = 0 (C) 9x2 - 8y2 - 18x - 9 = 0 (D) 9x2 - 8y2 + 18x + 9 = 0
5.
For hyperbola
6.
The line 2 x 6 y 2 touches the hyperbola x 2 2 y 2 4 at
x2 y2 1 , which of the following remains constant with cos 2 sin 2 change in ‘ ’ [IIT - 2003] (A) abscissa of vertices (B) abscissa of foci (C) eccentricity (D) directrix
(A) (4,- 6 ) 7.
(B) (2,-2 6 )
(C) (-4, 6 )
[IIT - 2004] (D) (-2,2 6 )
x 2 y2 1 and e2 is the eccentricity of the hyperbola passing If e1 is the eccentricity of the ellipse 16 25 through the foci of the ellipse and e1e2 = 1, then equation of the hyperbola is [IIT - 2006] x 2 y2 1 (A) 9 16
x 2 y2 1 (B) 16 9
x 2 y2 1 (C) 9 25
(D) none of these
Hyperbola
29
IIT JEE PROBLEMS
(SUBJECTIVE)
et et et et , y is a point on the hyperbola x 2 – y2 = 1. Show that the 2 2 area bounded by this hyperbola and the lines joining its centre to the points corresponding to t and – t1 is t1. [IIT - 82]
1.
For any real t, x
2.
A series of hyperbolas is drawn having a common transverse axis of length 2a. Prove that the locus of a point P on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is the curve (x2 - y2)2 = 4x2(x2 - a2). [REE-85]
3.
x2 y2 Let ‘p’ be the perpendicular distance from the centre C of the hyperbola 2 2 1 to the a b tangent drawn at a point R on the hyperbola. If S and S are the two foci of the hyperbola, then
FG H
2 2 show that ( RS RS) 4a 1
b2 p2
IJ K
[REE-89]
4.
Two straight lines pass through the fixed points ( a , 0) and have slopes whose products is p > 0. Show that the locus of the points of intersection of the lines is a hyperbola. [REE-93]
5.
Find the equations of the two tangents to the hyperbola xy = 27 which are perpendicular to the straight line 4x – 3y = 7. Also find the points of contact of these tangents. [REE-93]
6.
Determine the constant c such that the straight line joining the points (0, 3) and (5, –2) is tangent to the curve y = c/(x – 1). [REE-94]
7.
Determine the loci of the point which divides a chord with slope 4 of xy = 1 in the ratio 1 : 2. [IIT - 97]
8.
The angle between a pair of tangents drawn from a point P to the parabola y2 = 4ax is 45°. Show that the locus of the point P is a hyperbola. [IIT - 98]
9.
x 2 y2 1 to the circle x2 + y2 = 9. Find the Tangents are drawn from any point on the hyperbola 9 4 locus of midpoint of the chord of contact. [IIT - 2005]
30
Hyperbola
SET–I 1.
Equation of the hyperbola with eccentricity 3/2 and foci at (±2, 0) is (A)
2.
x2 y2 4 4 9 9
(B)
(B) (12) , 0
i
(B) latus rectum = 2 7
(C) e = 13 / 4
(D) none of these
The differential equation of all central conics whose axes are the axes of coordinates is
(C)
7.
8.
(D) 0, (12)
(A) foci are 13 , 0
2
6.
(D) none of these
(C) (13) , 0
d 2y dy (A) 1 + = 2 dx dx
5.
x2 y2 1 4 9
For the hyperbola whose transverse and conjugate axes are respectively 6 and 4 and whose centre is at the origin
d
4.
(C)
The foci of the hyperbola 4x2 - 9y2 - 36 = 0 are (A) (11) , 0
3.
x2 y2 4 9 4 9
dy 2 d 2 y y 2 y dx dx
=x
dy 2 d 2 y y 2 dx dx
(B) x dy dx
=y
dy dx
(D) none of these
If the eccentricity of the hyperbola x 2 y2 sec2 = 5 is ellipse x 2 sec2 + y2 = 25, then a value of is (A) /6 (B) /4 (C) /3 The line x cos y sin p touches the hyperbola
3 times the eccentricity of the
(D) /2
x2 y2 1 if a 2 b2
(A) a 2 cos2 b 2 sin 2 p 2
(B) a 2 cos2 b 2 sin 2 p
(B) a 2 cos2 b 2 sin 2 p 2
(D) a 2 cos2 b 2 sin 2 p
x2 y2 1 , then tangents Tangents drawn from a point on the circle x + y = 9 to the hyperbola 25 16 are at angle (A) /4 (B) /2 (C) /3 (D) 2 /3 2
2
The equation of the tangent lines to the hyperbola x 2 2y2 = 18 which are perpendicular to the line y = x are : (A) y = x ± 3 (B) y = x ± 3 (C)) 2x + 3y + 4 = 0 (D) none of these
Hyperbola
9.
31
For all real values of m, the straight line y = mx + (A) 9x 2 + 4y2 = 36 (C) 9x 2 4y2 = 36
10.
9 m 2 4 is a tangent to the curve
(B) 4x 2 + 9y2 = 36 (D) 4x 2 9y2 = 36
Identify the in correct statement(s) given below in respect of a hyperbola . x2 the asymptotes to this hyperbola 2 a
(A) (B)
y2 = 1 are the tangents from its centre . b2
if the eccentricity of the hyperbola is 5/4 then the eccentricity of its conjugate hyperbola will be 4/3 x2 y2 no pair of perpendicular tangents can be drawn to hyperbola = 1 from 4 16
(C)
its point . (D)
the AM of the slopes of the tangents to the hyperbola point (6 , 2) is 12/11 .
x2 y2 = 1 through the 25 16
11.
A point moves such that the sum of the squares of its distances from the two sides of length 'a' of a rectangle is twice the sum of the squares of its distances from the other two sides of length 'b' . The locus of the point can be (A) a circle (B) an ellipse (C) a hyperbola (D) none of these
12.
Equation of the hyperbola passing through the point (1, –1) and having asymptotes x + 2y + 3 = 0 and 3x + 4y + 5 = 0 is (A) 3x2 + 10xy + 8y2 + 14x + 22y + 7 = 0 (B) 3x2 – 10xy + 8y2 + 14x + 22y + 7 = 0 (C) 3x2 – 10xy + 8y2 – 14x + 22y + 7 = 0 (D) none of these
13.
An ellipse has eccentricity 1/2 and one focus at the point P (1/2, 1) . Its one directrix is the common tangent , nearer to the point P , to the circle x 2 + y2 = 1 and the hyperbola x2 y2 = 1. The equation of the ellipse in the standard form is (A)
(C)
14.
x 1 3
1
2
y 1 2 1 12
9
x 13 1
9
2
y 1 1 12
(B)
1
x 13 1
2
y 1
9
1 12
2
1
2
1
(D) none of these
x2 y2 x2 y2 1 1 are given to the confocal and length of and the hyperbola a 2 b2 A 2 B2 minor axis of ellipse is same as the conjugate axis of the hyperbola. If ee and eh represents the eccentricity The ellipse
1 1 2 is equal to 2 ee e h (C) 4 (D) 6
of ellipse and hyperbola respectively, then the value of (A) 1
(B) 2
32
15.
Hyperbola
Identify the incorrect statement(s) . 2
3 3x 4y 1 (A) the equation, (x 3) + (y + 2) = does not represent a hyperbola 2 5 2 2 x y (B) asymptotes to the hyperbola 2 2 = 1 are the tangents from the centre . a b 2
2
(C) difference of the focal distances of the point P(3 , 25/4) on the hyperbola, x2 y2 + 1 = 0 is 8 16 25
(D) none of these 16.
x2 y2 1 , subtends a right angle at a 2 2a 2 the origin. This chord will always touch a circle whose radius is A variable chord PQ, x cos y sin p of the hyperbola
(A) a
17.
(B)
a 2
(C) a 2
Which of the following equations in parametric form can represent a hyperbola, where 't' is a parameter . (A) x =
a b 1 1 t & y = t 2 2 t t
(C) x = et – et & y = et et 18.
(D) 2a 2
dx
(B)
tx y x ty +t=0 & + 1=0 a b a b
(D) none of these 3y
The differential equation dy = represents a family of hyperbolas (except when it 2x represents a pair of lines) with eccentricity (A)
3 5
(B)
5 3
(C)
2 5
(D) none of these
19.
Two parabolas y2 = 4a(x 1) & x 2 = 4a(y 2) always touch each other, 1 & 2 being variable parameters . Then their point of contact lies on a (A) straight line (B) circle (C) parabola (D) hyperbola
20.
Two conics a1x 2 + 2 h1xy + b1y2 = c1, a2x 2 + 2 h2xy + b2y2 = c2 intersect in 4 concyclic points . Then (A) (a1 b1) h2 = (a2 b2) h1 (B) (a1 b1) h1 = (a2 b2) h2 (C) (a1 + b1) h2 = (a2 + b2) h1 (D) (a1 + b1) h1 = (a2 + b2) h2
Hyperbola
33
SET–II 1.
x2 y2 For the hyperbola 2 2 = 1 the incorrect statement is a b
(A) the acute angle between its asymptotes is 60º (B) its eccentricity is 4/3 (C) length of the latus rectum is 2 (D) none of these 2.
The equation of common tangent to the curves y2 = 8x and xy = –1 is (A) 3y = 9x + 2 (B) y = 2x + 1 (C) 2y = x+8 (D) y = x+2
3.
The asymptotes of the hyperbola xy = hx + ky are (A) x k = 0 & y h = 0 (B) x + h = 0 & y + k = 0 (C) x k = 0 & y + h = 0 (D) x + k = 0 & y h = 0
4.
The locus of the centre of a circle which touches two given circles externally is (A) ellipse (B) parabola (C) hyperbola (D) None of these
5.
Area of the triangle formed by any tangent of the hyperbola xy = c2, and the coordinate axes, is equal to
6.
(A) 2c2
(B)
(C) 2 2 c
(D) 4c2
Total number of tangents of the hyperbola
x 2 y2 1 , that are perpendicular to the line 9 4
5x + 2y – 3 = 0, is/are (A) zero (C) 4 7.
8.
2c2
(B) 2 (D) none of these
Equations of a common tangent to the two hyperbolas
x2 y2 y2 x2 = 1 and = 1 is a 2 b2 a 2 b2
(A) y x a 2 b 2
(B) y x a 2 b 2
(C) y x a 2 b 2
(D) none of these
If the sum of the slopes of the normal from point P to the hyperbola xy = c2 is equal to ( R ) , then locus of point ‘P’ is (A) x 2 c2
(B) y2 c2
(C) xy = c 2
(D) none of these
34
9.
Hyperbola
Tangents at any point on the hyperbola
(C)
11.
12.
a2 x2 a2 y2
b2 y2 b2 x2
a2
b2
(B)
1
(D) none of these
x2
y2
1
Number of common tangent to the curves xy = c 2 & y2 = 4ax is (A) 0 (B) 1 (C) 2 (D) 4
Shortest distance between the curves
x2 a2
y2 b2
1 , 4x2 + 4y2 = a2 (b > a), is
(A)
b 2
(B)
b 2
(C)
a 2
(D)
a 2
A conic passes through the point (2, 4) and is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected at the point of tangency. Then the foci of the conic are
(C) (4, 4) & ( 4, 4)
4
2 & 4
(D)
(A) x = ab (C) y = b
, is 2
2
(B) 2 2 , 2 2 & 2 2 , 2 2 2,4
Locus of the point of intersection of tangents drawn to the curve sum of eccentric angles is
14.
1
(A) 2 2 , 0 & 2 2 , 0
13.
y2
1 cut the axes at A and B respectively. If the a 2 b2 rectangle OAPB (where O is origin) is completed then locus of point P is given by
(A)
10.
x2
x2 a2
y2 b2
2 , 4
1 at the points, whose
(B) y = ab (D) x = b
x2 y2 The co ordinates of a point on the hyperbola, = 1, which is nearest to the line 3x 24 18
+ 2y + 1 = 0 is (A) (3, 6) (C) (3, 3)
(B) (6 , 3) (D) none of these
Hyperbola
15.
35
The locus of the mid points of the chords passing through a fixed point (, ) of the hyperbola, x2 y2 2 = 1 is a2 b , 2 2
, 2 2
(A) a circle with centre
(B) an ellipse with centre
, 2 2
(C) a hyperbola with centre
16.
The asymptote of the hyperbola
, 2 2
(D) straight line passing through
x2 y2 = 1 form with any tangent to the hyperbola a a 2 b2
triangle whose area is a 2 tan in magnitude then its eccentricity is (A) sec (B) cosec (C) sec2 (D) cosec2 17.
The tangent to the hyperbola, x 2 3y2 = 3 at the point asymptotes constitutes (A) isosceles triangle (C) a triangles whose area is
18.
3 , 0 when associated with two
(B) an equilateral triangle 3 sq. units
(D) none of these
Latus rectum of the conic satisfying the differential equation, x dy + y dx = 0 and passing through the point (2, 8) is (A) 4 2
(B) 8
(C) 8 2
(D) 16
19.
The portion of the normal between the point P(x, y) of a curve & the x-axis is a constant. Then the curve is (A) a parabola (B) a circle (C) a rectangular hyperbola (D) none of these
20.
The locus of a point in the Argand plane that moves satisfying the equation, z 1 + i z 2 i = 3 (A) is a circle with radius 3 & centre at z = 3/2 (B) is an ellipse with its foci at 1 i and 2 + i and major axis = 3 (C) is a hyperbola with its foci at 1 i and 2 + i and its transverse axis = 3 (D) is none of the above
36
Hyperbola
SET–III Multiple choice question with one or more than one correct answer. 1.
The slopes of the common tangent to the hyperbolas (A) –2
2.
(B) –1
(C) 1
x 2 y2 y2 x 2 = 1 and = 1 are 9 16 9 16 (D) 2
x 2 y2 1 , parallel to the line y = x + 2, is The equation of the tangent to the hyperbola 4 3 (A) y = –x + 1 (B) y = x + 1 (C) y = –x – 1 (D) y = x – 1
3.
If the normals at (x i, yi) i = 1,2 , 3, 4 to the rectangular hyperbola xy = 2 meet at the point (3, 4), then (A) x 1 + x 2 + x 3 + x 4 = 3 (B) y1 + y2 + y3 + y4 = 4 (C) x 1 x 2 x 3 x 4 = –4 (D) y1 y2 y3 y4 = 4
4.
If the circle x2 + y2 = 1 cuts the rectangular hyperbola xy = 1 in four points (x i, yi), i = 1, 2, 3, 4 then (A) x 1 x 2 x 3 x 4 = –1 (B) y1 y2 y3 y4 = 1 (C) x 1 + x 2 + x 3 + x 4 = 0 (D) y1 + y2 + y3 + y4 = 0
5.
The equation
x 2 ( y 1) 2 x 2 ( y 1) 2 = K will represent a hyperbola for
(A) K (0, 2)
(B) K (0, 1)
(C) K (1, )
(D) K (0, )
Read the passage given below and answer the questions : The hyperbola which has BB as its transverse axis, and AA as its conjugate axis, is said to be the conjugate hyperbola of the hyperbola whose transverse and conjugate axes are respectively
y2 x 2 AA and BB . Thus the hyperbola 2 2 1 b a x 2 y2 1. a 2 b2 Two diameters are said to be conjugate when each bisects all chords parallel to the others. is conjugate to the hyperbola
b2 If y = mx, y = m1x be conjugate diameters, then mm1 2 . Let y = m1x + c be a set of chords a
6.
b2 parallel to y = m1x, then the diameter y = 2 x bisects them all. a m Through the positive vertex of the hyperbola at tangent is drawn; then it meet the conjugate hyperbola at
(A) a , b 2
(B)
2a , b
(C) (a, b)
(D) none of these
Hyperbola
7.
If e and e be the eccentricities of a hyperbola and its conjugate, then the value of (A) 2
8.
37
(B) 1
(C) 0
(D) none of these
1 1 2 is 2 e e
A straight line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate hyperbola in the points P and Q, then the tangents at P and Q meet on the curve
y 4 y 2 x 2 4x 2 (A) 4 2 2 2 b b a a
y 4 y 2 x 2 2x 2 (B) 4 2 2 2 b b a a
y 4 y 2 x 2 4x 2 (C) 4 2 2 2 b b a a
(D) none of these
9.
Tangents are drawn to a hyperbola from any point on one of the branches of the conjugate hyperbola, then their chord of contact will touch the (A) other branch of the conjugate hyperbola (B) at the point of contact (C) at the focus (D) none of these
10.
If the asymptotes are the straight lines x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and which passes through the points (1, –1), then the conjugate the hyperbola is (A) 3x 2 + 10xy + 9y2 + 14x + 22y + 23 = 0 (B) 3x 2 + 10xy + 8y2 + 14x + 22y + 23 = 0 (C) 3x 2 – 10xy + 8y2 + 14x + 22y + 23 = 0 (D) none of these Read the passage given below and answer the questions : If an incoming light ray passing through one focus S strike convex side of the hyperbola then it will get reflected towards other focus S .
11.
12.
A ray emanating from the point (5, 0) is incident on the hyperbola 9x 2 – 16y2 = 144 at the point P with abscissa 8, then the equation of the reflected ray after first reflection is (A) 3 3x 13 y 15 3 0
(B) 3 3x 13 y 15 3 0
(C) 3 3x 13 y 15 3 0
(D) none of these
In the above question (6) the point P lies in the first quadrant is
(C) 5, 3 3 (A) 8, 3
(B) 8, 3 3
(D) none of these
38
Hyperbola
13.
If a variable straight line x cos y sin p , which is a chord of the hyperbola
x 2 y2 1 (b a ) . Subtend a right angle at the centre of the hyperbola then it always touches a 2 b2 a fixed circle whose radius is (A) (C)
14.
ab (b 2a) ab 2
(b a 2 )
(B)
2ab (b 2 a 2 )
(D) none of these
PQ and QR are two focal chords of an ellipse and the eccentric angles of P, Q, R are 2, 2, 2 respectively. Then tan tan is equal to (A) cot (C) 2 cot
(B) cot 2 (D) none of these
Read the passage given below and answer the questions : Let the equation of the hyperbola be ax 2 + 2hxy + by2 + 2gx + 2fy + c = 0. Now the equation of the asymptotes differs from that of the hyperbola only by a constant, hence the equation of the asymptotes is ax 2 2 xhy by 2 2gx 2fy c 0 , where is to be so chosen that this may represent a pair of straight lines. Equation of the hyperbola differs from the equation of the asymptotes by the same constant that the equation of the asymptotes differs from that of the conjugate hyperbola , that is Hyperbola + Conjugate Hyperbola = 2 ( Pair of Asymptotes) while finding the asymptotes the value of can also be obtained by using the fact that the asymptotes pass through the centre. 15.
The pair of asymptotes of the hyperbola 6x 2 – 7xy – 3y2 – 2x – 8y – 6 = 0 are (A) 6x 2 – 7xy – 3y2 – 2x – 8y – 4 = 0 (B) 6x 2 – 7xy – 3y2 – 2x – 8y – 8 = 0 (C) 6x 2 – 7xy – 3y2 – 2x – 8y – 12 = 0 (D) 6x 2 – 7xy – 3y2 – 2x – 8y – 16 = 0
16.
The equation of the hyperbola, conjugate to the hyperbola given in above question is (A) 6x 2 – 7xy – 3y2 – 2x – 8y – 20 = 0 (B) 6x 2 – 7xy – 3y2 – 2x – 8y – 2 = 0 2 2 (C) 6x – 7xy – 3y – 2x – 8y – 14 = 0 (D) 6x 2 – 7xy – 3y2 – 2x – 8y – 10 = 0
17.
If asymptotes of hyperbola are x + 2y + 3 = 0 and 3x + 4y + 5 = 0 and hyperbola passes through (1, –1), then its equation is (A) (x + 2y + 3) (3x + 4y + 5) – 6 = 0 (B) (x + 2y + 3) (3x + 4y + 5) – 8 = 0 (C) (x + 2y + 3) (3x + 4y + 5) – 12 = 0 (D) (x + 2y + 3) (3x + 4y + 5) – 14 = 0
18. (i)
True or False The line 5x + 12y = 9 touches the hyperbola x 2 – 9y2 = 2 at the point (5, –4/3)
(ii)
The foci of the hyperbola (x 2/4) + (y2/12) = 1 coincide with the foci of the ellipse (x 2/25) + (y2/9) = 1.
Hyperbola
(iii)
39
If 3x 2 + 5xy + 2y2 = 0 is the pair of asymptotes of a hyperbola, then the pair of axes is
x 2 y 2 xy . 2 5 (iv)
The two concentric hyperbolas, whose axes meets at angles of 45º cut at right angle.
(v)
The equation 16x 2 – 3y2 – 32x – 12y – 44 = 0 represents a hyperbola with centre at (1, –2).
19.
Fill in the blanks : (i)
The coordinates of the foci of the hyperbola ______ .
(x 1) 2 (y 2) 2 = 1 are ______ and 9 16
(ii)
The equation to the locus of the feet of the perpendicular from the focus of the hyperbola 4x 2 9y2 = 36 upon any of its tangent has the equation________.
(iii)
The eccentricity of the hyperbola with its principal axes along the coordinate
axes and which passes through (3 , 0) and 3 2 , 2 is ______ . (iv)
If the foci of the ellipse
1 x2 y2 x2 y2 2 = 1 & the hyperbola = coincide then 25 25 b 144 81
the value of b 2 is ______ . (v) 20.
The foci of the hyperbola y2 x 2 = 1 has the coordinates ______ , ______ and ______ , ______ .
Match the column Column I (a) Eccentricity of rectangular hyperbola x 2 – y2 = a2, is
Column II (P) 6
( x 1) 2 ( y 1) 2 1, 9 16
(b)
If P is any point on the hyperbola
(Q)
(c)
and S1 and S2 are its foci, then |S 1P – S2P| is equal to The eccentricity of the conjugate hyperbola of hyperbola x 2 – 3y2 = 1 is The latus rectum of the hyperbola 9x 2 – 16y2 – 18x – 32y – 151 = 0 is A hyperbola passes through the points (3, 2) and (–17, 12) and has its centre at origin and transverse axis is along x-axis.
(R)
The length of its transverse axis is
(T)
(d) (e)
(S)
9 2
2 2
2 3
40
Hyperbola
LEVEL–I
ANSWER
1.
24x2 – 25y2 = 600
2.
3.
(–1, 2); (4, 2) & (–6, 2); 5x – 4 = 0 & 5x + 14 = 0;
4.
a
y2 – x2 = 5
32 , 3 6, 8; y – 2 = 0, x + 1 = 0, 4x – 3y + 10 = 0, 4x + 3y – 2 = 0
5 , b4 2
6.
x + y = 13
LEVEL–II 4. (x2 + y2)2 (a2y2 - b2x2) = x2y2(a2 + b2)2
9.
x 2 y2 1 4 2 4 a b a b2
IIT JEE PROBLEMS
(OBJECTIVE)
(A)
1.
FG x 1IJ H 3K by 1g
(B)
1.
ABCD
(C) 1.
D
2.
C
3.
B
4.
B
5.
B
6.
B
2
1 9
1 12
2
1
7.
B
Hyperbola
41
IIT JEE PROBLEMS 5.
7.
(SUBJECTIVE)
4y+3x = 36
16x2 + y2 + 10xy = 2
6.
c=4
9.
x 2 y2 x 2 y2 9 4 9
2
SET–I 1.
B
2.
D
3.
C
4.
B
5.
B
6.
D
7.
A
8.
B
9.
D
10.
B
11.
C
12.
A
13.
A
14.
B
15.
C
16.
B
17.
A
18.
B
19.
D
20.
A
SET–II 1.
B
2.
D
3.
A
4.
C
5.
A
6.
A
7.
A
8.
A
9.
A
10.
B
11.
C
12.
C
13.
C
14.
B
15.
B
16.
A
17.
C
18.
C
19.
B
20.
D
SET–III 1.
BC
2.
BD
3.
ABC
4.
BCD
5.
AB
6.
A
7.
B
8.
C
9.
A
10.
A
11.
A
12.
B
13.
A
14.
B
15.
A
16.
B
17.
B
18. (i) F
(ii)
T
(iii)
T
(iv)
T
(v)
T
19.(i) (–4, 2) and (6, 2)
(ii)
(v)
0, 2 and 0, 2
20.
a-R, b-P, c-T, d-Q, e-S
x2 + y2 = 9
(iii)
13 3
(iv)
b2 = 16
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