Conic Sections

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 - x + y + = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to

1. ( , 2 + 4 )
2. ( , 2 - 4 )
3. ( , 2 - 4 )
4. ( , 2 + 4 )

Let E1 : x29 + y24 = 1 be an ellipse. Ellipses Ei's are constructed such that their centres and eccentricities are same as that of E1, and the length of minor axis of Ei is the length of major axis of Ei+1 (i 1). If Ai is the area of the ellipse Ei, then 5 ( i=1 Ai ), is equal to .

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 .