Two concentric circles are of radii $5 c m$ and $3 c m$ . The length of the chord of the larger circle which touches the smaller circle is

4cm

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12cm

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2cm

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8cm

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Solution

The correct option is C 8cm
$G i v e n - O i s t h e c e n t r e o f t w o c o n c e n t r i c c i r c l e s$

$C_{1}, t h e i n n e r c i r c l e w i t h r a d i u s = O P = 3 c m$ and

$C_{2}, t h e o u t e r c i r c l e w i t h r a d i u s = O A = 5 c m .$

$A B i s a c h o r d o f C_{2} . I t t o u c h e s C_{1} a t P .$

$T o f i n d o u t -$

$T h e l e n g t h o f A B = ?$

$S o l u t i o n -$

$W e k n o w t h a t t h e r a d i u s t h r o u g h t h e p o i n t o f$ $c o n t a c t o f a t a n g e n t t o a c i r c l e i s p e r p e n d i c u l a r t o t h e t a n g e n t .$

$∴ O P ⊥ A B ⟹ ∠ O P A = 90^{o} . . . . . . . . . (i)$

$A g a i n w e k n o w t h a t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o a n y o f i t s c h o r d, b i s e c t s t h e l a t t e r .$

$∴ O P b i s e c t s A B a t P .$

$S o A B = 2 \times A P . . . . . . . (i i) .$
$N o w Δ O A P i s a r i g h t t r i a n g l e w i t h O A a s h y p o t e n u s e$

$(f r o m i) .$

$O A = 5 c m & O P = 3 c m .$

$S o, a p p l y i n g P y t h a g o r a s t h e o r e m, w e h a v e$

$A P = \sqrt{{O A}^{2} - {O P}^{2}} = \sqrt{5^{2} - 3^{2}} c m = 4 c m .$

$∴ A B = 2 \times A P (f r o m i i) = 2 \times 4 c m = 8 c m . A n s - O p t i o n D .$

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Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

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