Question

Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; then the points P, B, Q and T are not concyclic.

• A. True
• B. False

Answer 1

The correct option is B

False

Two circles intersect each other at point A and B. PAQ is a line which intesects circles at P, A and Q. At P and Q, tangents are drawn to the circles which meet at T.

Construction - Join AB, BP and BQ.
Proof - TP is the tangent and PA, a chord
$\therefore \angle TPA=\angle ABP\dots \left(i\right)\left(\text{angles in alt. segment}\right)$
Similarly we can prove that
$\angle TQA=\angle ABQ\dots \left(ii\right)$

Adding (i) and (ii) we get
$\angle TPA+\angle TQA=\angle ABP+\angle ABQ$

But in $\Delta PTQ,$
$\angle TPA+\angle TQA+\angle PTQ={180}^{\circ }$
$\Rightarrow \angle TPA+\angle TQA={180}^{\circ }-\angle PTQ$
$\Rightarrow \angle PBQ={180}^{\circ }-\angle PTQ$
$\Rightarrow \angle PBQ+\angle PTQ={180}^{\circ }$
But there are the opposite angles of the quadrilateral
$\therefore$ Quadrilateral PBQT is a cyclic

Hence P,B, Q and T are concyclic. Hence, the given statement is false.