Question

Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. Then, which of the following statements are true?

• A. $\angle PAQ=9\angle OPQ$
• B. $\angle PAQ=2\angle OPQ$
• C. $\angle OPQ=2\angle PAQ$
• D. $\angle PAQ=4\angle OPQ$

Answer 1

The correct option is B $\angle PAQ=2\angle OPQ$

Given - A circle with centre O. two tangents PA and QA are drawn from a point A outside the circle OP, OQ, OA and PQ are joined.

In quadrilateral OPAQ,
$\angle OPA=\angle OQA={90}^{\circ }$

$(\because OP\perp PAandOQ\perp QA)$

$\therefore \angle POQ+\angle PAQ+{90}^{\circ }+{90}^{\circ }={360}^{\circ }$

$\Rightarrow \angle POQ+\angle PAQ={360}^{\circ }-{180}^{\circ }={180}^{\circ }....\left(i\right)$

In $\Delta OPQ$, OP = OQ (radii of the circle)

$\therefore \angle OPQ=\angle OQP$

In $\Delta OPQ$, OP = OQ (radii of the circle)

$\therefore \angle OPQ=\angle OQP$

But $\angle POQ+\angle OPQ+\angle OQP={180}^{\circ }$

$\Rightarrow \angle POQ+\angle OPQ+\angle OPQ={180}^{\circ }$

$\Rightarrow \angle POQ+2\angle OPQ={180}^{\circ }$ ....(ii)

From (i) and (ii)

$\angle POQ+\angle PQQ=\angle POQ+2\angle OPQ$

$\angle PAQ=2\angle OPQ$