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

No worries! We‘ve got your back. Try BYJU‘S free classes today!

12cm

No worries! We‘ve got your back. Try BYJU‘S free classes today!

2cm

No worries! We‘ve got your back. Try BYJU‘S free classes today!

8cm

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

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

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

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

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

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

$S o l u t i o n -$

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

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

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

$∴ O P b i s e c t s A B a t P .$

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

$(f r o m i) .$

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

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

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

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

Suggest Corrections