Conic Sections

Let one root of the quadratic equation in x : (k2-15k+27 )x2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kx is equal to :

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

Let the directrix of the parabola P : y2=8x, cut x-axis at the point A. Let B( , ), >1, be a point on P such that the slope of AB is 3/5. If BC is a focal chord of P, then six times the area of ABC is :

1. 80
2. 160
3. 174
4. 192

Let the parabola y=x2+px+q passing through the point (1, -1) be such that the distance between its vertex and the x-axis is minimum. Then the value of p2+q2 is :

1. 2
2. 4
3. 5
4. 8

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 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

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 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 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 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

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

Let the focal chord PQ of the parabola y2 = 4x make an angle of 60 with the positive x axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y-axis at the point (0, ), then 5 2 is equal to:

1. 15
2. 25
3. 20
4. 30

Let the point P of the focal chord PQ of the parabola y2 = 16x be (1, -4). If the focus of the parabola divides the chord PQ in the ratio m : n, (m, n) = 1, then m2 + n2 is equal to :

1. 17
2. 37
3. 10
4. 26

Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :

1. 384
2. 392
3. 96
4. 192

Let y2=12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)= 1474. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2- x-64 3 y= , then - is equal to .

Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length 254 and it passes through the point (1, 0), then the area of ABCD is

1. 758
2. 1258
3. 252
4. 754

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B, and a denotes the area of SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is .

If the equation of the parabola with vertex V ( 32,3 ) and the directrix x+2y=0 is x2+ y2- xy-30x-60y+225=0, then + + is equal to :

1. 6
2. 8
3. 7
4. 9

If the line 3x-2y+12=0 intersects the parabola 4y=3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to

1. 2- -1 ( 32 )
2. -1 ( 97 )
3. -1 ( 119 )
4. -1 ( 45 )

Let P(4,4 3) be a point on the parabola y2=4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :

1. 34 33
2. 343 38
3. 17 3
4. 263 38

Let A, B and C be three points on the parabola y2 = 6x and let the line segment AB meet the line L through C parallel to the x-axis at the point D. Let M and N respectively be the feet of the perpendiculars from A and B on L. Then ( AM BNCD )2 is equal to .

Suppose AB is a focal chord of the parabola y2 = 12x of length l and slope m < 3. If the distance of the chord AB from the origin is d, then l d2 is equal to .

Let the length of the focal chord PQ of the parabola y2=12x be 15 units. If the distance of PQ from the origin is p, then 10p2 is equal to .