Write True or False and justify your answer in each of the following :

The length of tangent from an external point P on a circle with centre O is always less than OP.

True

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False

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Data insufficient

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Ambiguous

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Solution

The correct option is B False
$G i v e n - P A & P B a r e t a n g e n t s t o t h e c i r c l e w t t h c e n t r e O a t A & B r e s p e c t i v e l y . T o f i n d o u t - T h e a s s e r t i o n, P A o r P B i s a l w a y s > O A o r O B, i s t r u e o r f a l s e . J u s t i f i c a t i o n - P A & P B a r e t a n g e n t s t o t h e c i r c l e a t A & B r e s p e c t i v e l y . ∴ P A = P B a n d ∠ O A P & ∠ O B P a r e = 90^{o} ∴ Δ O A P i s a r i g h t o n e w i t h ∠ A = 90^{o} ⟹ ∠ A O P + ∠ A P O = 90^{o} (b y a n g l e s u m p r q p e r t y o f t r i a n g l e s) c a s e I - ∠ A O P = ∠ A P O ⟹ e a c h o f t h e m = 45^{o} . i . e P A = O A c a s e I I - ∠ A O P > ∠ A P O ⟹ ∠ A O P > 45^{o} & ∠ A P O < 45^{o} (i n a Δ t h e s i d e o p p o s i t e t o t h e g r e a t e r a n g l e i s g r e a t e r t h a n t h e s i d e o p p o s i t e t o t h e s m a l l e r a n g l e) ⟹ P A > O A c a s e I I I - ∠ A O P < ∠ A P O ⟹ ∠ A O P {< 45}^{o} & ∠ A P O > 45^{o} (s a m e a r g u m e n t a s c a s e I I) ⟹ P A > O A S o t h e a s s e r t i o n, P A o r P B i s a l w a y s < O A o r O B, i s f a l s e . A n s - F a l s e .$

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