Question

5. Prove that the perpendicular at the point of contact to the tangent to a circle passes
through the centre.
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Answer 1

Given a circle with center O and AB the tangent intersecting circle at point P

and prove that OP⊥AB

We know that tangent of the circle is perpendicular to radius at points of contact Hence

OP⊥AB

So, ∠OPB=90o..........(i)

Now lets assume some point X

Such that XP⊥AN

Hence ∠XPB=90o.........(ii)

From eq (i) & (ii)

∠OPB=∠XPB=90o

Which is possible only if line XP passes though O

Hence perpendicular to tangent passes though centre

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Answer 2

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