Conic Sections

An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the length of its latus rectum is :

1. 6
2. 6 3
3. 16 3
4. 4 3

Let A be the focus of the parabola y2 = 8x. Let the line y = mx + c intersect the parabola at two distinct points B and C. If the centroid of the triangle ABC is ( 73, 43 ), then (BC)2 is equal to :

1. 89
2. 80
3. 32
4. 41

Let the ellipse E : x2144 + y2169 = 1 and the hyperbola H : x216 - y2 2 = -1 have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H, then the value of 24(e + L) is :

1. 296
2. 126
3. 67
4. 148

Let a circle of radius 4 pass through the origin O, the points A(- 3a, 0) and B(0, - 2b), where a and b are real parameters and ab 0. Then the locus of the centroid of OAB is a circle of radius

1. 73
2. 113
3. 53
4. 83

Let each of the two ellipses E1: x2a2 + y2b2 = 1, (a > b) and E2: x2A2 + y2B2 = 1, (A < B) have eccentricity 45. Let the lengths of the latus recta of E1 and E2 be l1 and l2, respectively, such that 2l12 = 9l2. If the distance between the foci of E1 is 8, then the distance between the foci of E2 is

1. 965
2. 85
3. 165
4. 325

Let the image of parabola x2 = 4y, in the line x - y = 1 be (y + a)2 = b(x - c), a, b, c N. Then a + b + c is equal to

1. 12
2. 8
3. 6
4. 4

Let the length of the latus rectum of an ellipse x2a2 + y2b2 = 1, (a > b), be 30. If its eccentricity is the maximum value of the function f(t) = - 34 + 2t - t2, then (a2 + b2) is equal to

1. 276
2. 516
3. 256
4. 496

Let (h, k) lie on the circle C : x2 + y2 = 4 and the point (2h + 1, 3k + 2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to .

Let the domain of the function f(x) = 3 5 7 (9x - x2 - 13 ) be the interval (m, n). Let the hyperbola x2a2 - y2b2 = 1 have eccentricity n3 and the length of the latus rectum 8m3. Then b2 - a2 is equal to :

1. 7
2. 9
3. 11
4. 5

Let the line y - x = 1 intersect the ellipse x22 + y21 = 1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

1. - -1 ( 14 )
2. 2 + -1 ( 14 )
3. 2 + 2 -1 ( 14 )
4. 2 - -1 ( 14 )

Let PQ be a chord of the hyperbola x24 - y2b2 = 1, perpendicular to the x-axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3, then the area of the triangle OPQ is

1. 2 3
2. 115
3. 8 35
4. 95

If the points of intersection of the ellipses x2 + 2y2 - 6x - 12y + 23 = 0 and 4x2 + 2y2 - 20x - 12y + 35 = 0 lie on a circle of radius r and centre (a, b), then the value of ab + 18r2 is :

1. 53
2. 52
3. 55
4. 51

An equilateral triangle OAB is inscribed in the parabola y2 = 4x with the vertex O at the vertex of the parabola. Then the minimum distance of the circle having AB as a diameter from the origin is

1. 2(8 - 3 3)
2. 4(6 + 3)
3. 4(3 - 3)
4. 2(3 + 3)

If the line x + 2y = 1, where R, does not meet the hyperbola x2 - 9y2 = 9, then a possible value of is :

1. 0.6
2. 0.7
3. 0.8
4. 0.5

Let the set of all values of r, for which the circles (x+1)2 + (y+4)2 = r2 and x2 + y2 - 4x - 2y - 4 = 0 intersect at two distinct points be the interval ( , ). Then is equal to

1. 21
2. 24
3. 20
4. 25

If the chord joining the points P1(x1, y1) and P2(x2, y2) on the parabola y2 = 12x subtends a right angle at the vertex of the parabola, then x1x2 - y1y2 is equal to

1. 280
2. 288
3. 292
4. 284

Let the locus of the mid-point of the chord through the origin O of the parabola y2 = 4x be the curve S. Let P be any point on S. Then the locus of the point, which internally divides OP in the ratio 3 : 1, is :

1. 2x2 = 3y
2. 2y2 = 3x
3. 3y2 = 2x
4. 3x2 = 2y

Let S and S′ be the foci of the ellipse x225 + y29 = 1 and P( , ) be a point on the ellipse in the first quadrant. If (SP)2 + (S'P)2 - SP S'P = 37, then 2 + 2 is equal to :

1. 13
2. 15
3. 11
4. 17

Let P(10, 2 15) be a point on the hyperbola x2a2 - y2b2 = 1, whose foci are S and S'. If the length of its latus rectum is 8, then the square of the area of PSS' is equal to :

1. 4200
2. 1462
3. 900
4. 2700

Let PQ and MN be two straight lines touching the circle x2 + y2 - 4x - 6y - 3 = 0 at the points A and B respectively. Let O be the centre of the circle and AOB = /3. Then the locus of the point of intersection of the lines PQ and MN is :

1. x2 + y2 - 18x - 12y - 25 = 0
2. x2 + y2 - 12x - 18y - 25 = 0
3. 3(x2 + y2) - 12x - 18y - 25 = 0
4. 3(x2 + y2) - 18x - 12y + 25 = 0

Let the foci of a hyperbola coincide with the foci of the ellipse x236 + y216 = 1. If the eccentricity of the hyperbola is 5, then the length of its latus rectum is :

1. 96 5
2. 24 5
3. 12
4. 16

Let O be the vertex of the parabola x2 = 4y and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio 2 : 3 be the conic C. Then the equation of the chord of C, which is bisected at the point (1, 2), is :

1. 5x - 4y + 3 = 0
2. x - 2y + 3 = 0
3. 4x - 5y + 6 = 0
4. 5x - y - 3 = 0

If the line x + 4y = 7, where R, touches the ellipse 3x2 + 4y2 = 1 at the point P in the first quadrant, then one of the focal distances of P is :

1. 1 3 - 12 11
2. 1 3 - 12 5
3. 1 3 + 12 5
4. 1 3 + 12 7

Let y2 = 12x be the parabola with its vertex at O. Let P be a point on the parabola and A be a point on the x-axis such that OPA = 90 . Then the locus of the centroid of such triangles OPA is:

1. y2 - 4x + 8 = 0
2. y2 - 2x + 8 = 0
3. y2 - 9x + 6 = 0
4. y2 - 6x + 4 = 0

Let one end of a focal chord of the parabola y2 = 16x be (16, 16). If P( , ) divides this focal chord internally in the ratio 5 : 2, then the minimum value of + is equal to:

1. 5
2. 7
3. 22
4. 16

Let the ellipse 3x2 + py2 = 4 pass through the centre C of the circle x2 + y2 - 2x - 4y - 11 = 0 of radius r. Let f1, f2 be the focal distances of the point C on the ellipse. Then 6f1f2 - r is equal to

1. 78
2. 68
3. 70
4. 74

Let r be the radius of the circle, which touches x-axis at point (a, 0), a < 0 and the parabola y2 = 9x at the point (4, 6). Then r is equal to .

Let C1 be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C2 be the circle with centre (1, 3) that touches C1 externally at the point ( , ). If ( - )2 = mn, (m, n) = 1, then m + n is equal to

1. 22
2. 13
3. 9
4. 31

Let P be the parabola, whose focus is (-2, 1) and directrix is 2x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose abscissa is -2, is

1. 52
2. 32
3. 34
4. 14

Consider the hyperbola x2a2 - y2b2 = 1 having one of its foci at P(-3, 0). If the latus rectum through its other focus subtends a right angle at P and a2 b2 = 2 - , , N, then + is .