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Question

Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure.Then, the angles PAQ and PBQ are supplementary.



A

True

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B

False

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Solution

The correct option is A

True


Given - Two circles intersect each other at A and B A common tangent touches the circles at P and Q. PA, PB, QA and QB are joined.

Construction - Join AB,

Proof - PQ is the tangent and AB is the chord

QPA=PBA (alternate segment) ....(i)

similarly we can prove that

PQA=QBA (ii)

Adding (i ) and (ii), we get

QPA+PQA=PBA+QBA

But QPA+PQA=180PAQ= (iii)

(In ΔPAQ)

and PBA+QBA=PBQ (iv)

from (iii) and (iv)

PBQ=180PAQ

PBQ+PAQ=180

PAQ+PBQ=180

Hence PAQ and PBQ are supplementary.


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