Conic Sections

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