P is a point in the exterior of a circle having centre 0 and radius 24. OP = 25. A tangent from P touches the circle at Q. Find PQ.

P is in the exterior of a circle at distance 34 from the centre 0. A line through P touches the circle at Q. PQ = 16, find the diameter of the circle.

P is the point in the exterior of ⨀(O, r) and the tangents from P to the circle touch the circle at X and Y.

(1) Find OP, if r = 12, XP = 5

(2) Find m∠XPO, if m∠XOY = 110

(3) Find r, if OP = 25 and PY = 24

(4) Find m∠XOP, if m∠XPO = 80

and are the tangents drawn to ⨀ (O, r) from point P lying in the exterior of the circle and T and R are their points of contact respectively. Prove that m∠TPR = 2m∠OTR.

Prove that the perpendicular drawn to a tangent to the circle at the point of contact of the tangent passes through the centre of the circle.

In figure 11.24, two tangents are drawn to a circle from a point A which is in the exterior of the circle. The points of contact of the tangents are P and Q as shown in the figure. A line 1 touches the circle at R and intersects and in B and C respectively. If AB = c, BC = a, CA = b, then prove that

(1) AP + AQ = a + b + c

is a diameter of ⨀(O, 10). A tangent is drawn from B to ⨀(O, 8) which touches ⨀(O, 8) at D. intersects 0(0, 10) in C. Find AC.

Tangents from P, a point in the exterior of ⨀(O, r) touch the circle at A and B. Prove that and bisects.

A tangent from P, a point in the exterior of a circle, touches the circle at Q. If OP = 13, PQ = 5, then the diameter of the circle is

The points of contact of the tangents from an exterior point P to the circle with centre 0 are A and B. If m∠OPB = 30, then m∠ AOB = ……

A circle touches the sides , , of ΔABC at points D, E, F respectively. BD = x, CE = y, AF = z. Prove that the area of ΔABC = .