Conic Sections

Let x=9 be a directrix of an ellipse E, whose centre is at the origin and eccentricity is 13. Let P( , 0), >0, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :

1. 9y2=8x(1-x)
2. 3y2=4x(1-x)
3. 9y2=8x(x-1)
4. 3y2=4x(x-1)

Let P(3 , 2 ), 0, be a point on the ellipse x29+ y24=1, Q be a point on the circle x2+y2-14x-14y+82=0 and R be a point on the line x+y=5 such that the centroid of the triangle PQR is (2+ , 3+ 23 ). Then the sum of the ordinates of all possible points R is :

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

Let the circle x2 + y2 = 4 intersect x-axis at the points A(a, 0), a > 0 and B(b, 0). Let P(2 , 2 ), 0 < < 2 and Q(2 , 2 ) be two points such that ( - ) = 2. Then the point of intersection of AQ and BP lies on :

1. x2 + y2 - 4x - 4 = 0
2. x2 + y2 - 4x - 4y = 0
3. x2 + y2 - 4x - 4y - 4 = 0
4. x2 + y2 - 4y - 4 = 0

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 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 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

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 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

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

If A and B are the points of intersection of the circle x2 + y2 - 8x = 0 and the hyperbola x29 - y24 = 1 and a point P moves on the line 2x - 3y + 4 = 0, then the centroid of PAB lies on the line:

1. x + 9y = 36
2. 9x - 9y = 32
3. 4x - 9y = 12
4. 6x - 9y = 20

Let E: x2a2+ y2b2=1, a>b and H: x2A2- y2B2=1. Let the distance between the foci of E and the foci of H be 2 3. If a-A=2, and the ratio of the eccentricities of E and H is 13, then the sum of the lengths of their latus rectums is equal to :

1. 10
2. 7
3. 9
4. 8

Let P be a point on the ellipse x29+ y24=1. Let the line passing through P and parallel to y-axis meet the circle x2+y2=9 at point Q such that P and Q are on the same side of the x-axis. Then, the eccentricity of the locus of the point R on PQ such that PR: RQ=4: 3 as P moves on the ellipse, is :

1. 1321
2. 13923
3. 137
4. 1119

Let the locus of the midpoints of the chords of the circle x2+(y-1)2=1 drawn from the origin intersect the line x+y=1 at P and Q. Then, the length of PQ is :

1. 12
2. 1
3. 1 2
4. 2