#### Level – I

1. The abscissae of two points A and B are the roots of the equation x^{2} + 2ax – b^{2} = 0 and their ordinates are the roots of the equation x^{2} + 2px – q^{2} = 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( x^{2} – y^{2} ) = a^{2} – b^{2}.

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 x^{2} + y^{2} = r^{2} . 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 x^{2} + y^{2} = b^{2} generated by a point on the circle x^{2} + y^{2} = a^{2}touches the circle x^{2} + y^{2} = c^{2}, 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 x^{2} + y^{2} = 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 (m_{i} , 1/m_{i}) i = 1, 2, 3, 4 m_{i} > 0 are 4 distinct points on a circle,

then show that m_{1} .m_{2} .m_{3} .m_{4} = 1.

10. If 4α^{2} – 5β^{2} + 6α + 1 = 0 , Prove that αx + βy + 1 = 0 touches a definite circle. Find the centre and radius of the circle.