Question

In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.

Answer 1

From the given figure, we have

TP = TQ                              (Two tangents, drawn from an external point to a circle, have equal length.)

and

∠TQO=∠TPO=90°                 (Tangent to a circle is perpendicular to the radius through the point of contact.)

In ∆TOQ,

QT²+OQ²=OT²

⇒QT²=13²−5²=144

⇒QT=12 cm

Now,

OT − OE = ET = 13 − 5 = 8 cm

Let QB = x cm.

∴ QB = EB = x      (Two tangents, drawn from an external point to a circle, have equal length.)

Also, 

∠OEB = 90°        (Tangent to a circle is perpendicular to the radius through the point of contact.)

In ∆TEB,

EB²+ET²=TB²

⇒x²+8²=(12−x)²

⇒x²+64=144+x²−24x

⇒24x=80

`=>x = 80/24=10/3`

`:.AB=2x=20/3cm`

Thus, the length of AB is `20/3 cm`