N is the foot of the perpendicular form P to the diameter AB of the circle APB. P lies on the circle APB. Prove that `PB^2=AB.NB`,

AB is a diameter of a circle. C is any point on the circle, angleACB=

The tangent at a point P on an ellipse intersects the major axis at T ,a n dN is the foot of the perpendicular from P to the same axis. Show that the circle drawn on N T as diameter intersects the auxiliary circle orthogonally.

The quadrilateral ABCD is circumsribed about a circle. Prove that AB+ CD= BC+DA.

O is the centre of a centre having diameter AB. P is any point on the circle. Draw a perpendicular from P to AB, which meets AB at N. Prove that PB^2 =AB.BN

A B C is an equilateral triangle with A(0,0) and B(a ,0) , (a>0). L, M and N are the foot of the perpendiculars drawn from a point P to the side A B ,B C ,a n dC A , respectively. If P lies inside the triangle and satisfies the condition P L^2=P MdotP N , then find the locus of Pdot

AB is a diameter of a circle. PQ is such a chord of the circle that it is neither a diameter of the circle nor a interceptor of AB. By joining the points A, P and B, Q it is found that ABQP is a quadrilateral of wich angleBAP=angleABQ . Prove that ABQP is a cyclic trapezium.

PQ is a diameter of a circle and AB is such a chord of it that it is perpendicular to PQ . If C be the point of intersection of PQ and AB , then prove that PC.QC=AC.BC .

AB and CD are two diameters of a circle. Prove that ADBC is a rectangle

AB is a diameter of a circle. BP is a tangent to the circle at B . A straight line passing through A intersects BP at C and the circle at D . Prove that BC^(2)=ACxxCD .

Parama drew to circles intersect each other at the points P and Q . If the diameter of the two ircels are PA and PB resepctively , then prove that A,Q,B are collinear.