Q9 of 31 Page 233

is a chord of ⨀(O, 5) such that AB = 8. Tangents at A and B to the circle intersect in P. Find PA.

Given AB is a chord of circle (O, 5) such that AB = 8.


Let PR = x.


Since OP is a perpendicular bisector of AB,


AR = BR = 4



Consider ΔORA where ∠R = 90°,


By Pythagoras Theorem,


⇒ OA2 = OR2 + AR2


⇒ 52 = OR2 + 42


⇒ OR2 = 25 – 16 = 9


∴ OR = 3


⇒ OP = PR + RO = x + 3


Consider ΔARP where ∠R = 90°,


⇒ PA2 = AR2 + PR2


⇒ PA2 = 42 + x2 = 16 + x2 … (1)


Consider ΔOAP where ∠A = 90°,


⇒ PA2 = OP2 – OA2


= (x + 3)2 – (5)2


= x2 + 9 + 6x – 25


= x2 + 6x – 16 … (2)


From (1) and (2),


⇒ 16 + x2 = x2 + 6x – 16


⇒ 6x = 32


⇒ x =


From (1),


⇒ PA2 = 16 + x2


= 16 + () 2


= 16 +


=


=


∴ PA =


More from this chapter

All 31 →
7

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

 8

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.

 10

P lies in the exterior of ⨀(O, 5) such that OP = 13. Two tangents are drawn to the circle which touch the circle in A and B. Find AB.

 1

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 = .