Conic Sections

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 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

Let the length of a latus rectum of an ellipse x2a2 + y2b2 = 1 be 10. If its eccentricity is the minimum value of the function f(t) = t2 + t + 1112, t R, then a2 + b2 is equal to :

1. 115
2. 120
3. 125
4. 126

Let e1 and e2 be the eccentricities of the ellipse x2b2 + y225 = 1 and the hyperbola x216 - y2b2 = 1, respectively. If b < 5 and e1 e2 = 1, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

1. 45
2. 35
3. 74
4. 32

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the eccentricity of the ellipse x216 + y2n = 1 is :

1. 1 2
2. 12
3. 74
4. 34

The length of the latus-rectum of the ellipse, whose foci are (2, 5) and (2, -3) and eccentricity is 45, is

1. 503
2. 185
3. 65
4. 103

Let the sum of the focal distances of the point P(4, 3) on the hyperbola H: x2a2 - y2b2 = 1 be 8 53. If for H, the length of the latus rectum is l and the product of the focal distances of the point P is m, then 9l2 + 6m is equal to :

1. 187
2. 184
3. 186
4. 185

Let for two distinct values of p the lines y = x + p touch the ellipse E: x242 + y232 = 1 at the points A and B. Let the line y = x intersect E at the points C and D. Then the area of the quadrilateral ABCD is equal to :

1. 48
2. 20
3. 24
4. 36

The centre of a circle C is at the centre of the ellipse E: x2a2 + y2b2 = 1, a > b. Let C pass through the foci F1 and F2 of E such that the circle C and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle PF1F2 is 30 and the length of the major axis of E is 17, then the distance between the foci of E is :

1. 12
2. 26
3. 13
4. 132

The axis of a parabola is the line y=x and its vertex and focus are in the first quadrant at distances 2 and 2 2 units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is :

1. 8
2. 3
3. 9
4. 4

The radius of the smallest circle which touches the parabolas y = x2 + 2 and x = y2 + 2 is

1. 7 216
2. 7 28
3. 7 22
4. 7 24

A line passing through the point P( 5, 5) intersects the ellipse x236 + y225 = 1 at A and B such that (PA) (PB) is maximum. Then 5 (PA2 + PB2 ) is equal to :

1. 290
2. 377
3. 338
4. 218

Let C be the circle of minimum area enclosing the ellipse E : x2a2 + y2b2 = 1 with eccentricity 12 and foci ( 2, 0). Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 2a is parallel to the major axis of E and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is :

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