The distance of a chord AB of a circle from the centre is 8cm less than the radius, , If the radius of the circle is less than the length of the chord by 11cm then the respective lengths of the chord, the radius and the distance of chord from the centre are

13cm, 24cm, 5cm, and 29cm, 40cm, 21cm.

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24cm, 13cm, 5cm, and 40cm, 29cm, 21cm.

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24cm, 5cm, 13cm, and 40cm, 21cm, 29cm.

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5cm, 13cm, 24cm, and 21cm, 29cm, 40cm.

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Solution

The correct option is B 24cm, 13cm, 5cm, and 40cm, 29cm, 21cm.
$G i v e n - A B i s a c h o r d o f a c i r c l e w i t h c e n t r e O . O N i s t h e p e r p e n d i c u l a r d i s t a n c e o f A B f r o m O . T h e r a d i u s o f t h e c i r c l e = O A = r c m . O N = (r - 8) c m a n d A B = (r + 11) c m . T o f i n d o u t - A B = ? r = ? O N = ? S o l u t i o n - O N i s t h e p e r p e n d i c u l a r d i s t a n c e o f A B f r o m O . ∴ O N ⊥ A B i . e ∠ O N A = 90^{o} a n d A N = \frac{1}{2} \times A B = \frac{1}{2} \times (r + 11) c m s i n c e t h e p e r p e n d i c u l a r, d r o p p e d f r o m t h e c e n t e r o f a c i r c l e t o i t s a n y c h o r d, b i s e c t s t h e l a t t e r . N o w Δ O N A i s a r i g h t o n e w i t h O A = r c m a s h y p o t e n u s e s i n c e ∠ O N A = 90^{o} . S o, b y P y t h a g o r a s t h e o r e m, w e h a v e O A = \sqrt{{O N}^{2} + {A N}^{2}} ⟹ r = \sqrt{{(r - 8)}^{2} + {(\frac{1}{2} \times (r + 11))}^{2}} ⟹ r^{2} - 42 r + 377 = 0 ⟹ (r - 13) (r - 29) = 0 ⟹ r = (13, 29) c m . ∴ O N = (13 - 8) c m & (29 - 8) c m = (5, 21) c m . A n d A B = (13 + 11) c m & (29 + 11) c m = (24, 40) c m . ∴ T h e r e s p e c t i v e v a l u e s o f A B, r a n d O N a r e 24 c m, 13 c m 5 c m a n d 40 c m, 29 c m, 21 c m . A n s - O p t i o n B .$

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