Assertion (A): The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the centre. Reason (R): The tangent to a circle

Question

Assertion (A): The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the centre.

Reason (R): The tangent to a circle is perpendicular to the radius through the point of contact.

Smart Achievers · Accepted Answer

(1)

Let PA and PB be two tangents drawn from an external point P to the circle with centre O, as shown in Fig

$P A ⊥ O A and P B ⊥ O B$

$(T a n g e n t s a r e p e r p e n d i c u l a r t o t h e r a d i u s a t p o i n t o f c o n t a c t .)$

$I n q u a d r i l a t e r a l O A P B, w e h a v e :$

$∠ O A P + ∠ A P B + ∠ P B O + ∠ A O B = 360 °$

$\Rightarrow 90 ° + ∠ A P B + 90 ° + ∠ A O B = 360 °$

$\Rightarrow ∠ A P B + ∠ A O B = 180 °$

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Solution

Q.
Assertion (A): The angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact to the centre.

Reason (R): The tangent to a circle is perpendicular to the radius through the point of contact.

1 Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).

2 Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).

3 Assertion (A) is true but Reason (R) is false.

4 Assertion (A) is false but Reason (R) is true.

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