Conic Sections

If the midpoint of a chord of the ellipse x29 + y24 = 1 is ( 2, 4/3), and the length of the chord is 2 3, then is:

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

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 A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B, and a denotes the area of SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is .

Let the product of the focal distances of the point ( 3, 12 ) on the ellipse x2a2+ y2b2=1, (a>b), be 74. Then the absolute difference of the eccentricities of two such ellipses is

1. 1-2 2 3
2. 1- 3 2
3. 3-2 22 3
4. 3-2 23 2

Let circle C be the image of x2+y2-2x+4y-4=0 in the line 2x-3y+5=0 and A be the point on C such that OA is parallel to x-axis and A lies on the right hand side of the centre O of C. If B( , ), with <4, lies on C such that the length of the arc AB is (1/6)th of the perimeter of C, then - 3 is equal to

1. 4- 3
2. 3
3. 4
4. 3+ 3

If the equation of the parabola with vertex V ( 32,3 ) and the directrix x+2y=0 is x2+ y2- xy-30x-60y+225=0, then + + is equal to :

1. 6
2. 8
3. 7
4. 9

The equation of the chord, of the ellipse x225+ y216=1, whose mid-point is (3,1) is :

1. 5x+16y=31
2. 48x+25y=169
3. 4x+122y=134
4. 25x+101y=176

Let H1: x2a2- y2b2=1 and H2:- x2A2+ y2B2=1 be two hyperbolas having length of latus rectums 15 2 and 12 5 respectively. Let their eccentricities be e1= 52 and e2 respectively. If the product of the lengths of their transverse axes is 100 10, then 25e22 is equal to .

If the line 3x-2y+12=0 intersects the parabola 4y=3x2 at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to

1. 2- -1 ( 32 )
2. -1 ( 97 )
3. -1 ( 119 )
4. -1 ( 45 )

Let the circle C touch the line x-y+1=0, have the centre on the positive x-axis, and cut off a chord of length 4 13 along the line -3x+2y=1. Let H be the hyperbola x2 2- y2 2=1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then 2 2+3 2 is equal to .