Exercise 10.1: Circles — NCERT Solutions

A tangent to a circle touches it at exactly one point.

(a) Point inside the circle: Any line through an interior point cuts the circle at two points, so it is a secant. Hence no tangent can be drawn.

(b) Point on the circle: Exactly one tangent can be drawn at that point, perpendicular to the radius through the point.

(c) Point outside the circle: Exactly two tangents can be drawn.

Let $O$ be the centre of the circle and $XY$ a tangent touching the circle at point $P$.

We must prove $OP \perp XY$.

Take any point $Q$ on $XY$ other than $P$. Since $XY$ is a tangent, every point on it except $P$ lies outside the circle. Therefore $OQ > OP.$

This holds for every point $Q$ on $XY$ except $P$. So $OP$ is the shortest distance from $O$ to the line $XY$.

The shortest distance from a point to a line is the perpendicular distance. Hence $OP \perp XY.$

Thus the tangent is perpendicular to the radius at the point of contact.

Since $PQ$ is a tangent at $P$, the radius $OP \perp PQ$.

So triangle $OPQ$ is right-angled at $P$.

By the Pythagoras theorem: $OQ^2 = OP^2 + PQ^2$ $13^2 = 5^2 + PQ^2$ $169 = 25 + PQ^2$ $PQ^2 = 144$ $PQ = 12 \text{ cm}.$