Conic Sections

Let chord PQ of length 3 13 of the parabola y2=12x be such that the ordinates of points P and Q are in the ratio 1 : 2. If the chord PQ subtends an angle at the focus of the parabola, then is equal to :

1. 35
2. 45
3. 513
4. 1213

Let the centre of the circle x2+y2+2gx+2fy+25=0 be in the first quadrant and lie on the line 2x-y=4. Let the area of an equilateral triangle inscribed in the circle be 27 3. Then the square of the length of the chord of the circle on the line x=1 is .

Let C be a circle having centre in the first quadrant and touching the x-axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6 3 on y-axis, then the length of the chord of the circle C on the line x-y=3 is :

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

The eccentricity of an ellipse E with centre at the origin O is 32 and its directrices are x= 4 63. Let H: x2a2- y2b2=1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :

1. 4 2 7
2. 4 27
3. 4 7
4. 87

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 the line x-y=4 intersect the circle C: (x-4)2+(y+3)2=9 at the points Q and R. If P( , ) is a point on C such that PQ=PR, then (6 +8 )2 is equal to .

Let a focus of the ellipse E : x2a2+ y2b2=1 be S(4, 0) and its eccentricity be 45. If the point P(3, ) lies on E and O is the origin, then the area of POS is equal to:

1. 125
2. 145
3. 245
4. 485

Let P be a moving point on the circle x2+y2-6x-8y+21=0. Then, the maximum distance of P from the vertex of the parabola x2+6x+y+13=0 is equal to:

1. 8
2. 10
3. 12
4. 9

Let the point P be the vertex of the parabola y=x2-6x+12. If a line passing through the point P intersects the circle x2+y2-2x-4y+3=0 at the points R and S, then the maximum value of (PR+PS)2 is :

1. 10
2. 20
3. 25
4. 5

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