C and D are points on the circle with diameter AB such that $\angle AQB=2\angle COD,$ where O is the centre of the circle. Q is the point of intersection of AC and DB inside the circle. The tangents at C and D meet at P. The circle has radius 1. The distance of P from its center is

Q. The circle $OAB$ where $O$ is origin and $A,B$ are points on the co-ordinate axes is drawn such that the distances of points $A$ and $B$ from the tangent to the circle at origin are $p$ and $q$ respectively. Prove that the diameter of the circle is $p+q$ and its centre is $[\frac{1}{2}\sqrt{p(p+q)},\frac{1}{2}\sqrt{q(p+q)}]$.

Q.

From an external point P, two tangents are drawn that touch the circle at points Q and R. The centre of the circle is O. Points O and P are joined. The ratio of angles OPR and OPQ is

Q. Let two circles intersect each other at points A and D. Let the diameter AB intersect the circle with centre P at point N and the diameter AC intersect the circle at point M with centre Q. Then prove that AC. AM = AB. AN

Q.

AB is the diameter of the circle with centre O. P is a point on the circle such that $PA=2PB.$ If $AB=d,$ find BP.

Q. Three concentric circles with centre O and radii $r_1$ $r_2$ and $r_3$ are such that $r_1$, $r_2$, P is a point on the circle with radius $r_2$ and Q is a point on circle with radius r3 how many tangents can be drawn from P and Q to the circles?