AITS_MAT(PT_3)_NM
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NARAYANA IIT ACADEMY (Andhra Pradesh, Kota and Delhi) Max. Marks : 150
AITS – MATHEMATICS
Color : YELLOW Time: 2hrs.
Note : (1) Each question answered correctly is given +3 marks (2) Each question answered incorrectly is given 1 mark (3) 1 V questions are comprehensive containing 4 questions each
1.
A circle touches the y-axis and also touches the circle with center at (4, 0) and radius 3. The locus of the center of the circle is (a) a circle (b) an ellipse (c) a parabola (d) a hyperbola
2.
The locus of a point P(h, k) moving under the condition that the line y = kx + h is a tangent to the parabola y2 = 4ax is (a) a circle (b) an ellipse (c) a hyperbola (d) a parabola
3.
Let P(1, 4) and Q is a variable point on the lines |y| = 5, If PQ 3, then the number of positions of Q with integral coordinates is (a) 5 (b) 6 (c)8 (d) 7
4.
Let A(-3, 2), B(4, 3) and P(2a + 3, 2a - 3) is a variable point such that PA + PB is the minimum. Then ‘a’ is 41 (a) 0 (b) 12 39 (c) (d) 3 12
5.
Equation of the image of the lines y = |x – 1| by the line x = 3 is (a) y = |x – 5| (b) |y| = x – 5 (c) |y| + 4 = x (d) y = |x – 5
6.
If , , (< < ) are the points of discontinuity of the function f ( x )
1 , then the ln | x 1| values of ‘a’ for which the point (, ) and (, a2) lie on the same side of the line 4x – 3y + 4 = 0 are given by (a) (–, –2) (b) (–2, 2) (c) (2, ) (d) R
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AITS_MAT(PT_03) - 2
7
a, b, c, d, e, f, R and (a2+b2 + c2 + d2)x2 – 2(ab + bc + cd + de)x + (b2 + c2 + d2 + e2) 0, x R then the lines ax + by + c = 0 and cx + dy + e = 0 are (a) perpendicular (b) parallel (c) coincident (d) intersecting
8.
If the normal at three points (al2, 2al), (am2, 2am), (an2, 2an) of the parabola y2 = 4ax are concurrent, then the common roots of the equations lx2 + mx + n = 0 and a2(b2 – c2) x2 + b2(c2 – a2)x + c2(a2 – b2) = 0 is (a) independent of a, b, c but dependent on l, m, n (b) independent of l,m,n but dependent on a,b, c (c) independent of a, b, n but dependent on c, l, m (d) independent of a, b, c, l, m, n,
9.
If f(x) =
10.
The locus of the foot of the perpendicular from the origin to chords of the circle x2 + y2 – 2x – 4y – 4 = 0 which subtend a right angle at the origin is (a) x2 + y2 – x –2y – 2 = 0 (b) 2(x2 + y2) –2x – 4y + 3 = 0 2 2 (c) x + y – 2x – 4y + 4 = 0 (d) x2 + y2 + x + 2y – 2 = 0
11.
If x – 3y + 1 = 0 is a diameter of a circle circumscribing the rectangle ABCD. A is (–2, 5) and B is (6, 5). Then which of the statement is not true (a) Area of the rectangle ABCD is 64 sq. units (b) Centre of the circle is (2, 1) (c) The other two vertices of the rectangle are (–2, –3) and (6, –3) (d) Equation of the sides are x = –2, y = –3, x = 5 and y = 6
12.
If the focus of parabola (y – b) 2 = 8(x – a) always lies between the line x + y = 2 and x + y = 6, then a + b lies in interval (a) (0, 2) (b) (0, 1) (c) (0, 4) (d) (1, 4)
13.
The straight line y = µx + (µ > 0) touches the parabola y2 = 16 (x + 4), then the minimum value taken by ‘’ is. (a) 4 (b) 8 (c) 16 (d) 32
14.
If two distinct chords are drawn from the point (4, 2) on the parabola x 2 = 8y are bisected on the line y = mx, then the set of values of ‘m’ is given by
4 and f(x) (, ], then a ray of light parallel to the axis of the parabola 2 x2 y2 = 8x, after reflection from the internal surface of the parabola will necessarily pass through the point (a) (, ) (b) (, ) (c) (, ) (d) (, )
(a) , 2 1 (c) (0, ) 15.
2 1,
(b) R
(d) 2 1, 2 1
A chord of slope ‘m’ of the circle x2 + y2 = 2 touches the parabola y2 = 8x, then set of values of m is given by NARAYANA IIT ACADEMY South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
AITS_MAT(PT_03) - 3
(a) (, ) (c) (0, ) 16.
(b) (–1, 1) (d) (, 1) (1, )
The point on x + y = 2, which is nearest to the parabola y2 = 4(x – 6) is 11 7
7
11
11
(b) , 2 2
(a) , 2 2 7
(c) , 2 2
7
11
(d) , 2 1
17.
If the area of the triangle with vertices at (, ), (, ) and (, ) is A, then the area of the triangle with vertices at (αγ β 2 ,αβ γ 2 ),(αβ , 2 ) and (β , 2 ) is (a) A (b) (α + β + γ) A 2 (c) (α + β + γ) A (d) (α + β + γ)3 A
18.
If O is the origin and OA, OB are the tangents from the origin to the circle x2 + y2 – 4x + 2y + 4 = 0, the circumcentre of the triangle OAB is
1
1
(a) 1, 2
(c) 1, 2
19.
1
(b) 1, 2 1
(d) 1, 2
The range of values of ‘b’ such that the angle between the tangents drawn from
(0, b) to the circle x2 + y2 = 4 lies in , is (where is the angle which contains the circle) 3 2 (a) (c)
2, 4 4, 2 2 2
(b) 2 2, 4 2, 4
(d) (–4, 4)
20.
The equation of the chord of the circle x 2 + y2 – 6x + 2y + c = 0 passing through the point (1, 2) farthest from the center is (a) 2x – 3y – 4 = 0 (b) 2x + 3y + 4 = 0 (c) 2x + 3y – 4 = 0 (d) 2x – 3y + 4 = 0
21.
If
22.
A curve y = f(x) passes through the point (6, 8) and the normal to the curve at that point happens to be a tangent to the circle x2 + y2 = 100. The value of f '(6) is
(a 2b) 2 3 c (a 2b) 0 , the family of lines ax + by + c = 0 is concurrent at the point 6c 2 1 2 1 2 (a) , (b) , 3 3 3 3 1 2 1 2 (c) , (d) , 3 3 3 3
3 4 4 (c) 3
(a)
4 3 3 (d) 4
(b)
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AITS_MAT(PT_03) - 4
23.
24.
If the straight lines (m + 1)x + ay + 1 = 0, (m + 2)x + by + 1 = 0, and (m + 3)x + cy + 1 =0, m 0, are concurrent, then a, b, c are in (a) A.P. only for m = 1 (b) H.P. for all m (c) G.P. for all m (d) A.P. for all m Vertices of a variable triangle are (5, 12), (–13 cos, 13 sin ), and (13 sin , – 13 cos ). Then locus of its orthocentre is (a) x – y + 7 = 0 (b) x – y – 7 = 0 (c) x + y – 7 = 0 (d) x + y + 7 = 0
25.
A straight line L with positive slope passes through the points (4, –4) and cuts the co-ordinate axes in fourth quadrants at points A and B. As L varies, the absolute minimum value of OA + OB is (0 is origin) (a) 0 (b) 16 (c) 8 (d) 18
26.
The equation (x 4) 2 (y 1) 2 (x 4) 2 (y 1) 2 8 represents (a) a circle (b) a pair of straight lines (c) an ellipse with foci (± 4, 0) (d) a line segments joining points (–4, –1) and (4, –1)
27.
If the vertices A and B of a triangle ABC are given by (1, 3), and (2, –7) and C moves along the line L : 3x + 2y + 1 = 0, the locus of the centroid of the triangle ABC is a straight line parallel to (a) AB (b) BC (c) CA (d) ‘L’
28.
If the parabola y = ax2 + 2cx + b touch the x-axis. Then the line ax + by +c = 0 (a) passes through a fixed point (b) has a fixed direction (c) makes a triangle with co-ordinate axes of constant area (d) data insufficient
29.
Centroid of a triangle is origin. Circumcentre of the triangle is the fixed point from where all the lines ax + by + c = 0 pass, where a, b, c are distinct non-zero real parameters following the relation a3 + b3 + c3 = 3abc. The orthocentre of the triangle will be
1 1
(a) , 2 2
(b) (-1, -1)
(c) (-2, -2)
(d) (-2, 2)
30.
A point is taken on the director circle of the circle x2 + y2 = 8 and tangents are drawn from it to the parabola y2 = 16x. the locus of the middle points of chord of contact will be (a) y2 + (y2 – 8x)2 = 1024 (b) 64y2 + (y2 – 8x)2 = 1024 2 2 2 (c) y – (y – 8x) = 1024 (d) 64y2 – (y2 – 8x)2 = 1024
P-1.
Find the point P, on the circle x 2 + y2 – 6x – 8y + 16 = 0, where O is the origin and OX is the positive side of the x-axis.
31.
POX is minimum NARAYANA IIT ACADEMY South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
AITS_MAT(PT_03) - 5
28 96
32.
96 28
, (a) 25 25
, (b) 25 25
(c) (0, 4)
(d)
OP is maximum 8 24
8 32
(a) , 5 5
(b) , 5 5
24 32
32 24
, (c) 5 5
33.
, (d) 5 5
OP is minimum 4 32
8 6
(a) , 5 5
(b) , 5 5
6 8
32 24
(c) , 5 5 34.
96 28 , 25 25
, (d) 5 5
POX is maximum 28 96
, (b) 25 25
(a) (0, 4) 24 32
8 6
, (c) 5 5
(d) , 5 5
P-2.
Consider two concentric circle C1 : x2 + y2 – 4 = 0 and C2 : x2 + y2 – 9 = 0. A parabola is drawn through the points where ‘C1’ meets the y-axis and having arbitrary tangent of ‘C2’ as its directrix. ‘C’ is the curve of the locus of the focus of drawn parabola
35.
The curve ‘C’ is (a) Circle (c) Ellipse
(b) Parabola (d) Hyperbola
36.
The number of common tangents to the circle x2 + y2 – 6x – 8y – 24 = 0 and C2 is (a) 3 (b) 2 (c) 1 (d) 0
37.
If the centroid of an equilateral triangle is the center of the curve ‘C’ and its one vertex is (1, 1), then the equation of its circumcircle is (a) x2 + y2 = 4 (b) x2 + y2 = 1 (c) x2 + y2 = 3 (d) x2 + y2 = 2
38.
The point (, ) lies inside the curve ‘C’. Then, 2 lies in interval
45
45 , 14 45 (d) 1, 14
(a) 0, 14
(b)
(c) (0, ) P-3.
cos 2n x; If 1 lim n
x n ,
n I and
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AITS_MAT(PT_03) - 6
1 lim sin 2n x; where x (2n 1) , n I n 2
Then the point P(, ) is translated parallel to the line x – y = 4 by a distance 4 2 units, if the new position Q is in the third quadrant, then 39.
40.
The co-ordinate of P(, ) is (a) (2, 1) (c) (2, 2)
(b) (1, 2) (d) (0, 2)
The co-ordinates of Q is (a) (–2, –4) (c) (–2, –3)
(b) (–2, –1) (d) (–3, –2)
41.
The locus of the point A such that PAQ is a right angle. (a) x2 + y2 – 2y + 7 = 0 (b) x2 + y2 + 2y – 7 = 0 2 2 (c) x + y + 2y + 7 = 0 (d) x2 + y2 – 2y – 7 = 0
42.
Mr. Mishra starts walking from the point B (–3, 3) and will reach the point C(O, 4) touching the line PQ at D. so that he will travel in the shortest distance, then ‘D’ is 5
(a) ,1 3 2 5
(c) , 3 3
5 2
(b) , 3 3 5 2
, (d) 3 3
P-4.
The parabola described parameterically by x = sin 2t, y = sin t + cos t, then
43.
A line L passing through the focus of the parabola intersects the parabola in two distinct points. If ‘m’ be the slope of the line L, then (a) m < –1 or m > 1 (b) –1 < m < 1 (c) m < 2 (d) m R ~ {0} The circle x2 + y2 + 2kx = 0, kR. touches the directrix and latus-rectum of the parabola, then the set of the values of k is given by 3 5 3 (a) , (b) , 4 4 4 3 5 3 (c) , (d) , 4 4 4
44.
45.
The locus of the point whose chord of contact w.r.t the parabola passes through the focus of the parabola is (a) x + 5/4 = 0 (b) x + 6/5 = 0 (c) x + 8/5 = 0 (d) x + 2 = 0
46.
If and be the length of the line segments of focal chord of parabola, then (a) 2
(b) 4
1 (c) 2
(d)
1 1 is
1 4
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AITS_MAT(PT_03) - 7
P-5.
Suppose f(x, y) = 0 is the equation of the circle such that f(x, –2) = 0 has equal roots (each equal to 1) and f(2, x) = 0 also has equal roots (each equal to –1), then
47.
The center of the circle is (a) (1, 1) (c) (–1, –1)
48.
(b) (–1, 1) (d) (1, –1)
Area of an equilateral triangle inscribed in the circle f(x, y) = 0 is 3 3 sq. units 4 9 (c) sq. units 4
(a)
3 sq. units 4 3 (d) sq. units 4
(b)
49.
The equation of the smallest circle passing through the intersection of the line x – y = 1 and the circle f(x, y) = 0 (a) x2 + y2 + x + y = 0 (b) x2 + y2 – x + y = 0 (c) x2 + y2 – x – y = 0 (d) x2 + y2 + x – y = 0
50.
If the tangent at the point A on the circle f(x, y) = 0 meets the straight line 3x + 2y – 6 = 0 at a point B on the x-axis, then the length of AB is (a) 2 (b) 2 2 (c) 1 (d) 2 3
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AITS_MAT(PT_03) - 8
NARAYANA IIT ACADEMY Color : YELLOW
(Andhra Pradesh, Kota and Delhi)
AITS – MATHEMATICS ANSWERS 1
2
3
4
5
6
7
8
9
10
C
C
A
B
A
B
C
D
C
A
11
12
13
14
15
16
17
18
19
20
D
C
B
A
D
C
C
A
C
D
21
22
23
24
25
26
27
28
29
30
A
B
D
A
B
D
D
C
C
B
31
32
33
34
35
36
37
38
39
40
B
C
C
A
C
B
D
A
A
C
41
42
43
44
45
46
47
48
49
50
B
B
D
C
A
B
D
A
B
C
Hints and Solution 1.
CC1 = h + 3 (h – 4)2 + k2 = (h + 3)2 k2 – 8h + 16 = 6h + 9 k2 – 14h + 7 = 0
h
C (h , k )
h
1 Required locus of C(h, k) is y2 = 14 x 2 2.
3.
3
C 1
(4 , 0 )
a m a h hk = a k hence, locus of P(h, k) is a hyperbola c
Q = (x, 5), so PQ 3 (x – 1)2 + (4 m 5)2 9 1 + (x –1)2 9 [Q 81 + (x –1)2 9 is absurd] (x – 1)2 8 2 2 x 1 2 2 1 – 2 2 x 1 + 2 2 , but x is an integer. x = –1, 0, 1, 2, 3
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AITS_MAT(PT_03) - 9
4.
PA + PB AB (by triangle inequality) so, PA + PB is minimum if PA + PB = AB i.e. A, P, B are collinear 3 2 1 4 3 1 0 2a 3 2 a 3 1 –3(3 – 2a +3) + 2(2a + 3 – 4) + 1(8a – 12 – 6a – 9) = 0 –18 + 6a + 4a – 2 + 2a – 21 = 0 12a – 41 = 0 a = 41/12
5.
x =3 -1
)
-(x
(x
y=
=(
x–
5)
)
(x
Y
-5
– Y=
y=
–1 )
( 1 ,0 )
( 3 ,0 )
(5 ,0 )
6.
Points = 0, = 1 and = 2 (0, 1) and (2, a2) are same side of line 4x – 3y + 4 = 0 12 – 3a2 > 0 a2 < 4 i.e. – 2 < a < 2
7.
(ax – b)2 + (bx – c)2 + (cx – d)2 + (dx – e)2 0 a, b, c, d, e are in G.P.
8.
1 is the common root of the both the equations.
9.
0
4 2 2 x2
Focus of y2 = 8x is (2, 0) 10.
Equation to the chord is xx1 + yy1 = x12 y12 Where M(x1, y1) is a foot of or from the origin homogenize the equation to the circle given, as we get A 2 2 2 2 2 2 2 2 x y (x + y ) x1 y1 - (2x + 4y) (xx1 + yy1). 1 1 – 4(xx1 + yy1) = 0. This represents a pair of perpendicular lines through the origin i.e. M (x , y ) 1
2 x12 y12 x12 y12 (2x1 4y1 ) 4 x12 y12 0
1
2
giving the locus of (x1, y1) is 2(x2 + y2) – 2x – 4y – 4 = 0 i.e. x2 + y2 – x – 2y – 2 = 0 11.
C
90º (0 , 0 )
(1 , 2 )
Area of the rectangle = 8 × 8 = 64 sq. units
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B
AITS_MAT(PT_03) - 10
x = 2 (6 , – 3 ) C
(– 2 , – 3 ) D H
A (– 2 , 5 )
(2 , 1 )
M (2 , 5 )
x – 3y + 1 = 0
B (6 , 5 )
12.
Co-ordinates of focus will be S(a + 2, b), S lies to the opposite sides of origin w.r.t. line x + y – 2 = 0 and same side as origin w.r.t line x + y – 6 = 0 hence, 0 < a + b < 4
13.
Tangent to
y2 = 16(x + 4) is y = µ(x + 4) +
c = 4µ + 14.
4
c 8
Hence, the minimum value of c is 8 Any point on the line y = mx can be taken (t, mt). Equation of chord of the parabola with (t, mt) as mid-point is x.t – 4(y + mt) = t2 – 8mt It passes through (4, 2), 4t – 4(2 + mt) = t2 – 8mt t2 – 4(m + 1)t + 8 = 0 D>0 For two distinct chords 16(m + 1)2 – 4.8 > 0
m , 2 1
15.
4
2 1,
Equation of tangent to the parabola with slope ‘m’ is y mx
2 m
This line to be chord of the circle x2 + y2 = 2, If
2 m
2
4 < (m4 + m2).2
1 m2
m4 + m2 – 2 > 0 m2 –1 > 0 i.e. m (, 1) (,1) 16.
Any point on x + y = 2 is (, 2 – ). A lines through (, 2 - ) perpendicular to x + y = 2 is y – (2 – ) = x – x – y = 2 – 2 ….(1) 2 (1) should be a normal to y = 4(x – 6). Equation of normal is y + t(x – 6) = 2t + t3 …..(2) from (1) & (2) 11
11 2 7
Hence point is , 2 2
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AITS_MAT(PT_03) - 11
17.
18.
2 1 2 2 2
2 1 1 2 2 1 1 ( ) 1 ( ) 2 A 2 2 1 1
Equation of chord of contact AB, is 2x – y - 4 = 0 Required Equation of circle is x2 + y2 – 4x + 2y + 4 + (2x – y - 4) = 0 (1) passes through (0, 0) ….. = 1
…..(1)
Equation of circumcircle is x2 + y2 – 2x + y = 0
19.
20.
/
2 2 b 6 2 4 1 2 1 2 |b| 2
(0 , b )
2
P
sin
A 2
2 2 | b | 4
i.e.
O B
(CM)2 = (CP)2 – (PM)2 which is maximum when PM is minimum i.e. coincides with M (i.e. middle point of chord) Hence equation is 2x - 3y + 4 = 0
C P (1 , 2 ) M
21.
(a – 2b + 3c)2 = 0
22.
As (6, 8) lies on the circle, the normal to y = f(x) is the tangent to the circle at (6, 8), so that circle intersect orthogonally at (6, 8) 4 f '(6) = – the reciprocal of the slope of the circle at (6, 8) = 3
23. 24.
m+1 a 1 m+2 b 1 = 0 m+3 c 1
2b = a + c
Circumcentre is (0, 0) 5 13cos 13sin 12 13sin 13cos , 3 3
Centroid G is
G divides HO internally 2 : 1
5 13cos 13sin 2.0 1.h 3 3 12 13sin 13cos 2.0 1.k 3 3
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AITS_MAT(PT_03) - 12
h 5 13 k 12 sin cos 13 h 5 k 12 from (1) and (2), 13 13
sin cos
…..(1) …..(2)
h–k+7=0 Locus of orthocentre H(h, k) is x–y+7=0 25.
Equation of line is y + 4 = m(x – 4), as m > 0 mx – y = 4m + 4 x y 1 4m 4 (4m 4) m 4m 4 4m 4 OA OB m 4 4m 8 16 m
27.
Let (h, k) be the centroid of ABC with C as (, ) 1 2 3 3 7 & k 3 h
= 3h – 3 = 3k + 4
since C(, ) lies on L. 3(3h – 3) + 2(3k + 4) + 1 = 0 9h + 6k = 0 Locus of (h, k) is 3x + 2y = 0 which is parallel to ‘L’. 28.
Solving equation of parabola with x-axis (i.e. y = 0). we get ax2 + 2cx + b = 0, which should have two equal values of x, as x-axis touch the parabola. B (0 , -c /b ) discriminant = 0 4c2 – 4ab = 0 c2 = ab
29.
A
O
1 c c now area of 2 a b 1 c2 2 ab 1 now c2 = ab, so area = 2
(0 , -c /a )
ax + by + c = 0
a3 + b3 + c3 = 3abc a + b + c = 0 as a2 + b2 + c2 ab + bc + ca (a, b, c distinct) Now ax + by +c = 0 is equation of line and on comparing with the relation x = 1, y = 1. so circumcentre is (1, 1). Centroid is (0, 0), let orthocentre H (x1, y1) ( x 1, y 1)
H
2 G
(0 , 0 )
C
(1 , 1 )
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AITS_MAT(PT_03) - 13
0
x1 2 , 3
0
2 y1 3
x1 = -2 y1 = -2 so orthocentre is (-2, -2) 30.
Director circle of the given circle x2 + y2 = a2 will be x2 + y2 = a 2
2
x2 + y2 = 16 now point on it will be (4 cos, 4sin) Now chord of contact for the parabola y2 = 16x, T = 0 y 4 sin = 8(x + 4 cos ) . . . . . (i) equation of the chord whose middle point is (h, k). T = S1 ky – 8(x + h) = k2 – 16h 8x – ky + k2 – 8h = 0 . . . . . (ii) comparing the coefficient of the two equation of the same line k sin , 4
31.
k 2 8h cos , so 64y2 + (y2 – 8x)2 = 1024 32
Q
OC = 5, CP = 3, OP = 4, C (3, 4) OL = 3 OM = OL + LM = OL + HP 4cos = 3 + 3sin 4 – 3 tan = 3 sec 16 + 9 tan2 – 24 tan = 9 sec2 tan = 7/24
(3 , 4 )
H
96 28 , 25 25
P (OP cos , OP sin )
32.
O
Max. value of OP = OQ = OC + radius = 8, m OC
C
N
L
P
X
K M
4 tan 3
P (OQ cos , OQ sin ) 24 32 , 5 5 P (2cos , 2sin ) P
33.
3 4 P 2 ,2 5 5
6 8 , 5 5
34.
P(0, 4)
P-2.
Let the focus of parabola be (x1, y1) SP = distance of P from directrix Here P are (0, 2) and (0, –2) Eq. Of directrix is x cos + y sin = 3 x12 (y1 2) 2 (2sin 3) 2 and x2 + (y1 + 2)2 = (2 sin + 3)2 Subtract, we get 8y1 = 24 sin sin = y1/3 Adding, (1) and (2), 2(x12 y12 4) 2(4sin 2 9)
(0 , 2 ) ….. …..
(3 c
(1) (2)
os
,
(0 , 2 ) 3 s in )
NARAYANA IIT ACADEMY South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
AITS_MAT(PT_03) - 14
y12 9 9 9x12 9y12 36 4y12 81
9x12 5y12 45
x12 y12 4 4
36.
x 2 y2 1 5 9 Distance between their centers = 5 |r1 – r2| < 5 < r1 + r2 Hence, number of common tangents = 2
37.
(x – 0)2 + (y – 0)2 =
38.
39.
lim(1 cos 2n x), x n , n I n
35.
Locus of focus is
2
2
142 – 45 < 0 45 0 2 14
=1+1=2
2
sin 2n x, x (2n 1) , n I and 1 lim n
=1+0=1 40.
1 1 Q 2 4 2 ,1 4 2 2 2
(–2, –3) 41.
A (h , k )
PA2 + AQ2 = PQ2 Locus of A(h, k) is x2 + y2 + 2y – 7 = 0
90º
Q (– 2 , – 3 )
P (2 , 1 ) B ( 4 , – 4 )
42.
Equation of PQ is y = x – 1 Image of B(–3, 3) w.r.t line y = x – 1 is B(4, –4) For the shortest distance BDC are collinear. Equation of BC is O y + 2x = 4
D 5 ,2 3 3
43.
D
Equation of the parabola is y2 = x + 1
……(1)
3 focus is , 0 4
y = x – 1
C (0 , 4 ) B (– 3 , 3 )
Any line through the focus is 3 y 0 m x 4
…..(2) NARAYANA IIT ACADEMY
South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
AITS_MAT(PT_03) - 15
Solving (1) and (2) 2
3 m x (x 1) 4 9 3 m 2 x 2 m 2 1 x m 2 1 0 16 2 2
If m 0 D>0
2
9 4 9 4 3 2 2 9 2 2 2 m m 1 4m 1 m 3m 1 m 4m 4 4 2 16
= m2 + 1 > 0 for all m But if m = 0, then x does not have two real values distinct. m R , except m = 0
44.
3 5 k 4 4
(0 , 0 ) (– k , 0 )
45.
x = – 5 /4
Equation of chord of contact from the point P(h, k) is ky
hx 1 2
x = – 3 /4
…..(1)
3
(1) passes through focus ,0 4 5 4
Required locus is x 0
46.
Semi-latus-rectum is the harmonic mean of line segments of focus chords i.e.
1 1 4
P-5
47. 48.
Equation of circle f(x, y) = 0 is x2 + y2 – 2x + 2y + 1 = 0 C (1, –1) 3 3 2 2 3 area of an equilateral 3 4 3 3 sq. units 4
P
BC 2CQ cos30 2 1
49.
C (1 ,– 1 ) Q
30º
R
Equation of circle is x2 + y2 – 2x + 2y + 1 + (x – y – 1) = 0 x2 + y2 + ( – 2)x + (2 – )y + 1 – = 0 For smallest circle x – y = 1 is a diameter of the required circle
2 ( 2) 1 2 2
=1 NARAYANA IIT ACADEMY
South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
AITS_MAT(PT_03) - 16
Hence circle is x2 + y2 – x + y = 0 50.
Point B is (2, 0). Length of tangent = s1 AB 22 0 4 1 1
NARAYANA IIT ACADEMY South Centre: 47 B, Kalu Sarai, N. Delhi-16. North West Centre: 2/40, Adj. Central Mkt., W. Punjabi Bagh, N. Delhi-26.
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