Conic Sections

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

If one of the diameters of the circle x2+y2-10x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of the lines 2x+3y=12 and 3x-2y=5, then the radius of the circle C is :

1. 4
2. 3 2
3. 6
4. 20

If the foci of a hyperbola are same as that of the ellipse x29+ y225=1 and the eccentricity of the hyperbola is 158 times the eccentricity of the ellipse, then the smaller focal distance of the point ( 2, 143 25 ) on the hyperbola, is equal to

1. 14 25- 43
2. 7 25+ 83
3. 7 25- 83
4. 14 25- 163

Let the foci and length of the latus rectum of an ellipse x2a2+ y2b2=1, a>b be ( 5, 0) and 50, respectively. Then, the square of the eccentricity of the hyperbola x2b2- y2a2b2=1 equals

Let P be a parabola with vertex (2,3) and directrix 2x+y=6. Let an ellipse E: x2a2+ y2b2=1, a>b, of eccentricity 1 2 pass through the focus of the parabola P. Then, the square of the length of the latus rectum of E, is

1. 51225
2. 65625
3. 3858
4. 3478

If the circles (x+1)2+(y+2)2=r2 and x2+y2-4x-4y+4=0 intersect at exactly two distinct points, then

1. 12<r<7
2. 3<r<7
3. 5<r<9
4. 0<r<7

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

1. 1 3
2. 2 5
3. 32
4. 53

Let the latus rectum of the hyperbola x29- y2b2=1 subtend an angle of 3 at the centre of the hyperbola. If b2 is equal to lm(1+ n), where l and m are co-prime numbers, then l2+m2+n2 is equal to .

Let P be a point on the hyperbola H: x29- y24=1, in the first quadrant such that the area of triangle formed by P and the two foci of H is 2 13. Then, the square of the distance of P from the origin is

1. 26
2. 22
3. 20
4. 18