A pair of tangents can be constructed to a circle inclined at an angle of $170^{0} .$

True

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False

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Ambiguous

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Data Insufficient

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Solution

The correct option is A True
$L e t u s d r a w t w o t a n g e n t s f r o m P t o t h e c i r c l e w i t h c e n t r e O a t Q & R . O Q & O R a r e j o i n e d . W e h a v e O Q & O R a s r a d i i o f t h e c i r c l e . A n d ∠ O Q P & ∠ O R P e a c h = 90^{o} s i n c e P Q & P R a r e t a n g e n t s t o t h e c i r c l e a t Q & R r e s p e c t i v e l y . ∴ ∠ O Q P + ∠ O R P = 180^{o} . . . . . . . . . . (i) B u t ∠ Q P R + ∠ Q O R + (∠ O Q P + ∠ O R P) = 360^{o} (A n g l e s u m p r o p e r t y o f q u a d r i l a t e r a l s) ⟹ ∠ Q P R + ∠ Q O R + 180^{o} = 360^{o} (f r o m i) ⟹ ∠ Q P R + ∠ Q O R = 180^{o} S o e a c h o f ∠ Q P R & ∠ Q O R < 180^{o} . T h e g i v e n a n g l e i s 170^{o} w h i c h i s l e s s t h a n 180^{o} . S o, t h e s t a t e m e n t, t h a t a p a i r o f t a n g e n t s c a n b e d r a w n t o a c r c l e i n c l i n e d a t a n a n g l e o f 170^{o}, i s t r u e . A n s - t r u e$

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Question 4
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[Thinking process :
If the angle between a pair of tangents is greater than $180^{\circ}$ , then the pair of tangent can never be constructed to a circle.]

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A pair of tangents can be constructed to a circle inclined at an angle of 1700.

A pair of tangents can be constructed to a circle inclined at an angle of $170^{0} .$