SSC CGL 20 Aug 2021 Shift 1 Paper

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Question : 51 of 100

Marks: +1, -0

In a circle with centre

O

and radius

13 {cm}

, a chord

A B

is drawn. Tangents at

A

and

B

intersect at

P

such that

∠ A P B=60^{∘}

. If distance of

A B

from the centre

O

is

5 {cm}

, then what is the length (in

{cm}

) of

A P

?

12
11
22
24

Solution:

Given,

∠ A P B=60^{∘}

Radius

=13 {cm}

In

△ {AOM}

,

⇒(A O)^{2}=(O M)^{2}+(M A)^{2}

⇒(13)^{2}=(5)^{2}+(M A)^{2}

⇒ 169-25=(M A)^{2} ⇒ 144=(M A)^{2}

⇒ \;\; M A=12 {cm}

In

△ A P B

,

∠ A P B+∠ P A B+∠ A B P=180^{∘}

[ ∵ A P=P B

∴ ∠ P A B=∠ P B A]

\;⇒\;2 ∠ P A B\;=180^{∘}-60^{∘}

⇒\;2 ∠ P A B\;=120^{∘}

⇒\;∠ P A B\;=60^{∘}

\;\;\text " Now, "\; \cos 60^{∘}\;=\;{\;\text " Base "\;}/{\;\text " Hypotenuse "\;}=\;{A M}/{P A}

⇒\;\;{1}/{2}\;=\;{12}/{A P}

\;A P\;=24 {cm}

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