Question

Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

Answer 1

STEP :Proof

Let us assume a circle with centre $O$ and let $AB$ be the tangent intersecting the circle at point $P$.

Also let us assume a point $X$ such that $XP$ is perpendicular to $AB$.

STEP 2 : Proving that $XP$ passes through centre $O$

We know that

Tangent of a circle is perpendicular to radius at point of contact

$\Rightarrow OP\perp AB$ (Tangent at any point of circle is perpendicular to the radius through point of contact)

So, $\angle OPB=90°$ $...\left(1\right)$

we have already assumed that $XP$ is perpendicular to $AB$

$\angle XPB=90°$ $...\left(2\right)$

Now from equation $\left(1\right)$ and $\left(2\right)$

$\angle OPB=\angle XPB=90°$

This condition is possible only if line $XP$ passes through $O$.

Since, $XP$ passes through centre $O$.

Therefore, it is proved that the perpendicular at the point of contact to the tangent of a circle passes through the centre.