Geometry (C10)

In figure, AB is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at A and B intersect at P. Find the length of AP.

Answer

AB = 8 cm

⇒ AM = 4 cm

∴ OM = √(52 – 42) = 3 cm

Let AP = y cm, PM = x cm

∴ ∆OPA is a right angle triangle

∴ OP2 = OA2 + AP2

(x + 3)2 = y2 + 25

⇒ x2 + 9 + 6x = y2 + 25 ...(i)

Also, x2 + 42 = y2 ...(ii)

⇒ x2 + 6x + 9 = x2 + 16 + 25

⇒ 6x = 32

⇒ x = 32/6 = 16/3 cm

∴ y2 = x2 + 16 = 256/9 + 16 = 400/9

⇒ y = 20/3 cm