Conic Sections

Let O be the vertex of the parabola y2=4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is :

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

Let x2f (a2+7a+3 )+ y2f(3a+15)=1 represent an ellipse with major axis along y-axis, where f is a strictly decreasing positive function on R. If the set of all possible values of a is R-[ , ], then 2+ 2 is equal to :

1. 28
2. 40
3. 61
4. 24

Consider the circle C : x2+y2-6x-8y-11=0. Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord AB is the circle x2+y2- x- y- =0, then + +2 is equal to .

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

If the eccentricity e of the hyperbola x2a2- y2b2=1, passing through (6, 4 3 ), satisfies 15 (e2+1 )=34e, then the length of the latus rectum of the hyperbola x2b2- y22 (a2+1 )=1 is :

1. 10
2. 20
3. 25
4. 30

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)