Conic Sections

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 set of all values of r, for which the circles (x+1)2 + (y+4)2 = r2 and x2 + y2 - 4x - 2y - 4 = 0 intersect at two distinct points be the interval ( , ). Then is equal to

1. 21
2. 24
3. 20
4. 25

Let C1 be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C2 be the circle with centre (1, 3) that touches C1 externally at the point ( , ). If ( - )2 = mn, (m, n) = 1, then m + n is equal to

1. 22
2. 13
3. 9
4. 31

If the four distinct points (4, 6), (-1, 5), (0, 0) and (k, 3k) lie on a circle of radius r, then 10k + r2 is equal to

1. 34
2. 32
3. 35
4. 33

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

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 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 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

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2,5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval ( , ), then 3 -2 is equal to :

1. 10
2. 12
3. 14
4. 15

Let the parabola y=x2+px-3, meet the coordinate axes at the points P, Q and R. If the circle C with centre at (-1,-1) passes through the points P, Q and R, then the area of PQR is :

1. 4
2. 6
3. 5
4. 7

Let a circle passing through (2, 0) have its centre at the point (h, k). Let (xc, yc ) be the point of intersection of the lines 3x + 5y = 1 and (2 + c)x + 5c2y = 1. If h = c 1 xc and k = c 1 yc, then the equation of the circle is :

1. 5x2 + 5y2 - 4x - 2y - 12 = 0
2. 25x2 + 25y2 - 20x + 2y - 60 = 0
3. 25x2 + 25y2 - 2x + 2y - 60 = 0
4. 5x2 + 5y2 - 4x + 2y - 12 = 0

Let the circles C1 : (x - )2 + (y - )2 = r12 and C2 : (x - 8)2 + (y - 152 )2 = r22 touch each other externally at the point (6, 6). If the point (6, 6) divides the line segment joining the centres of the circles C1 and C2 internally in the ratio 2 : 1, then ( + ) + 4 (r12 + r22 ) equals

1. 130
2. 110
3. 145
4. 125

A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n2 is equal to

1. 408
2. 414
3. 312
4. 396

Let C be the circle of minimum area touching the parabola y = 6 - x2 and the lines y = 3|x|. Then, which one of the following points lies on the circle C ?

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

Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :

1. 4 2
2. 4
3. 5
4. 2 2

Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC. Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies :

1. r = 1
2. 2r2 - 4r + 1 = 0
3. 2r2 - 8r + 7 = 0
4. r2 - 8r + 8 = 0

A square is inscribed in the circle x2+y2-10x-6y+30=0. One side of this square is parallel to y=x+3. If (xi, yi ) are the vertices of the square, then (xi2+yi2 ) is equal to:

1. 152
2. 148
3. 156
4. 160

Let C be a circle with radius 10 units and centre at the origin. Let the line x + y = 2 intersect the circle C at the points P and Q. Let MN be a chord of C of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord MN is

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

Let C: x2+y2=4 and C': x2+y2-4 x+9=0 be two circles. If the set of all values of so that the circles C and C' intersect at two distinct points, is R-[a, b], then the point (8a+12, 16b-20) lies on the curve :

1. x2+2y2-5x+6y=3
2. 5x2-y=-11
3. x2-4y2=7
4. 6x2+y2=42

If one of the diameters of the circle x2+y2-10x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of the lines 2x+3y=12 and 3x-2y=5, then the radius of the circle C is :

1. 4
2. 3 2
3. 6
4. 20

If the circles (x+1)2+(y+2)2=r2 and x2+y2-4x-4y+4=0 intersect at exactly two distinct points, then

1. 12<r<7
2. 3<r<7
3. 5<r<9
4. 0<r<7

Four distinct points (2k,3k),(1,0),(0,1) and (0,0) lie on a circle for k equal to :

1. 313
2. 213
3. 513
4. 113