There are two concentric circles, each with centre $O$ and if radii $10 c m$ and $26 c m$ respectively. The length of the chord $A B$ of the outer circle which touches the inner circle at $P$ is

48cm

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24cm

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40cm

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20cm

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Solution

The correct option is A 48cm
$G i v e n - O i s t h e c e n t r e o f t w o c o n c e n t r i c c i r c l e s a n d A B, w h i c h i s a c h o r d o f t h e o u t e r c i r c l e, t o u c h e s t h e i n n e r c i r c l e a t P . T h e r a d i u s o f t h e i n n e r c i r c l e i s 10 c m a n d t h e r a d i u s o f t h e o u t e r c i r c l e i s 26 c m . T o f i n d o u t - t h e l e n g t h o f A B = ? S o l u t i o n - W e j o i n O P . T h e n O P i s a r a d i u s t h r o u g h P w h i c h i s t h e p o i n t o f c o n t a c t o f A B t o t h e i n n e r c i r c l e . i . e P O = 10 c m . A l s o w e j o i n O A . T h e n O A i s t h e r a d i u s o f t h e o u t e r c i r c l e = 26 c m . N o w O P ⊥ A B ⟹ ∠ O P A = 90^{o} s i n c e t h e r a d i u s o f a c i r c l e t h r o u g h t h e p o i n t o f c o n t a c t o f a t a n g e n t t o t h e c i r c l e i s p e r p e d i c u l a r t o t h e t a n g e n t . ∴ Δ O P A i s a r i g h t o n e w i t h O A a s h y p o t e n u s e . S o, a p p l y i n g P y t h a g o r a s t h e o r e m, A P = \sqrt{{O A}^{2} - {O P}^{2}} = \sqrt{26^{2} - 10^{2}} c m = 24 c m B u t A B = 2 A P s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o a n y o f i t s c h o r d, b i s e c t s t h e l a t t e r . i . e P i s t h e m i d p o i n t o f A B . ∴ A B = 2 \times 24 c m = 48 c m . A n s - O p t i o n A .$

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There are two concentric circles, each with centre O and if radii 10cm and 26cm respectively. The length of the chord AB of the outer circle which touches the inner circle at P is

There are two concentric circles, each with centre $O$ and if radii $10 c m$ and $26 c m$ respectively. The length of the chord $A B$ of the outer circle which touches the inner circle at $P$ is