If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

y = sqrt(3)x +lambda is drawn through focus S of the parabola y^(2)= 8x +16 . If two intersection points of the given line and the parabola are A and B such that perpendicular bisector of AB intersects the x-axis at P then length of PS is

Two circles with centres O and O' of radii 3cm and 4 cm, respectively intersect at two points P and Q, such that OP and O' P are tangents to the two circles. Find the length of the common chord PQ.

Two circles intersect at A and B. From a point P on one of the circles lines PAC and PBD are drawn intersecting the second circle at C and D. Prove that CD is parallel to the tangent at P.

A straight line moves so that the product of the length of the perpendiculars on it from two fixed points is constant. Prove that the locus of the feet of the perpendiculars from each of these points upon the straight line is a unique circle.

Perpendiculars are drawn, respectively, from the points Pa n dQ to the chords of contact of the points Qa n dP with respect to a circle. Prove that the ratio of the lengths of perpendiculars is equal to the ratio of the distances of the points Pa n dQ from the center of the circles.

Prove by vertor metord thtat if a line is drawn frome the centre of a circle of a circle to the midpoint of a chord then the line is perpendicular to the chord

In the adjoining figure, two circles intersect each other at points A and E . Their common secant through E intersects the circle at points B and D . The tangents of the circles at point B and D intersect each other at point C . Prove that squareABCD is cyclic .