Conic Sections

Let the line x + y = 1 meet the circle x2 + y2 = 4 at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

1. 14
2. 3 7
3. 2 14
4. 5 7

If x + y = 109 is the equation of the chord of the ellipse x29 + y24 = 1, whose mid point is ( 52, 12 ). then + is equal to :

1. 37
2. 46
3. 72
4. 58

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

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

Let y2=12x be the parabola and S be its focus. Let PQ be a focal chord of the parabola such that (SP)(SQ)= 1474. Let C be the circle described taking PQ as a diameter. If the equation of a circle C is 64x2+64y2- x-64 3 y= , then - is equal to .

Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length 254 and it passes through the point (1, 0), then the area of ABCD is

1. 758
2. 1258
3. 252
4. 754

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 .