Level – I
1. The abscissae of two points A and B are the roots of the equation x2 + 2ax – b2 = 0 and their ordinates are the roots of the equation x2 + 2px – q2 = 0. Find the equation and the radius of the circle with AB as diameter.
2. Two rods of length a and b slide along the axes, which are rectangular, in such a manner that their ends are concyclic. Prove that the locus of the centres of the circle passing through these ends is the curve 4( x2 – y2 ) = a2 – b2.
3. Suppose f(x , y) = 0 is the equation of the circle such that f(x , 1) = 0 has equal roots (each equal to 2) and f(1 , x) = 0 also has equal roots (each equal to 0). Find the equation of the circle.
4. A circle touches the line 2x + 3y + 1 = 0 at the point (1, -1) and is orthogonal to the circle which has the line segment having end points (0, -1) and (-2, 3) as the diameter.
5. Two tangents are drawn from the point P (6 , 8) to the circle x2 + y2 = r2 . Find r such that the area of the triangle formed by the tangents and the chord of contact is maximum.
6. If the chord of contact of the circle x2 + y2 = b2 generated by a point on the circle x2 + y2 = a2touches the circle x2 + y2 = c2, prove that a, b, c are in G.P.
7. Find the equations of the tangents from the point A(3 , 2) to the circle x2 + y2 = 4 and hence find the angle between the pair of tangents.
8. A variable circle passes through the point A (a, b) and touches the x – axis. Show that the locus of the other end of the diameter through A is (x – a)2 = 4by.
9. If (mi , 1/mi) i = 1, 2, 3, 4 mi > 0 are 4 distinct points on a circle,
then show that m1 .m2 .m3 .m4 = 1.
10. If 4α2 – 5β2 + 6α + 1 = 0 , Prove that αx + βy + 1 = 0 touches a definite circle. Find the centre and radius of the circle.