Circles C(O, r) and C(O', r'), `(r gt r') ` touch internally at P. PQ is a chord of circle C(O, r) which intersects C(O', r') at R. Show that OO'RQ is a trapezium

Two circles C(O, r) and C'(O', r') touch externally at P(3, 1). If the co-ordinates of O and O' are (1, p) and (q, -2) repectively. Their areas are in the ratio 4 : 9. Find the value of p^(2)+q^(2) .

P Q and R S are two parallel chords of a circle and lines R P and S Q intersect each other at O as shown in Figure. Prove that O P=O Qdot

Diagonals A C and B D of a quadrilateral A B C D intersect at O in such a way that a r(triangle A O D)=a r(triangle B O C) . Prove that A B C D is a trapezium.

From a point P, two tangents PA and PB are drawn to a circle with centre O and radius r. If OP=2r, show that triangleAPB is equilateral.

If O is the centre of a circle with radius r and AB is a chord of the circle at a distance r/2 from O, then /_BAO=

P Q\ a n d\ R S are two parallel chords of a circle and lines R P\ a n d\ S Q intersect each other at O as shown in Figure. Prove that O P=O Q

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot

Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that a r(A O D)= a r(B O C)dot Prove that ABCD is a trapezium.

Diagonals A C\ a n d\ B D of a quadrilateral A B C D intersect at O in such a way that a r\ ( A O D)=a r\ (\ B O C)dot Prove that A B C D is a trapezium.

Diagonals A C\ a n d\ B D of a trapezium A B C D with A B||D C intersect each other at O . Prove that a r\ ( A O D)=\ a r\ ( B O C) .