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Q.

In figure, O is the center 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.

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a

5.3 cm

b

3.3 cm

c

4.3 cm

d

6.3 cm

answer is A.

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Detailed Solution

Given that O is the center of a circle of radius 5 cm. T is a point such that

OT = 13 cm

and

OT

intersects circle at E.
We have to find the length of AB.
Question Image

Question Image

By Pythagoras theorem, for a right-angled triangle ,

A B C, A C 2 = A B 2 + B C 2

, where AC is the hypotenuse of

△ A B C

Given: O is the center of a circle of radius 5cm. T is a point such that OT = 13cm and OT intersects circle at E. If AB is a tangent to the circle at E. To Find: Find the length of AB where TP and TQ are two tangents to the circle.
From the right angle triangle POT,

PT= OT 2 -OP 2 ⇒ PT= 132 -5 2 ⇒ PT= 169-25 ⇒ PT= 144

So, PT=12 cm.
As the radius of the circle is 5 cm,

ET=OT-OE ⇒ E T=13-5 ⇒ E T=8 cm

Suppose PA = AE = x.
In the right angle triangle AET,

TA 2 =TE 2 +EA 2 ⇒ (12-x) 2 =64+x 2 ⇒ 144-24x+x 2 =64+x 2 ⇒ 24x=80

So,

x= 8024 =3 .3 cm

The length of AB = 3.3 cm.
Therefore, the correct option is 1.

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