Question

Two circles touch each other externally at point P. Q is a point on the common tangent through P. Show that the tangents drawn from Q to the given two circles are equal in length.

Answer 1

Two circles touch each other externally at point P.

The common tangent at the point of contact P touches both circles at P.

Let Q be any point on this common tangent.

1. Tangents from Q to the first circle

From point Q, QA and QP are tangents to the first circle.

By the tangent–tangent theorem:

QA = QP …(i)

Tangents from Q to the second circle.

From point Q, QB and QP are tangents to the second circle.

Therefore:

QB = QP …(ii)

3. Comparing (i) and (ii)

Since both equal QP, we get:

QA = QB