Question

prove that the penpendicular at the point of contact to the tangent to a circle passes through the centre

Answer 1

Let AB be the tangent to the circle at point P with centre O.

We have to prove that PQ passes through the point O.

Suppose that PQ doesn't passes through point O. Join OP.

Through O, draw a straight line CD parallel to the tangent AB.

PQ intersect CD at R and also intersect AB at P.

AS, CD // AB PQ is the line of intersection,

∠ORP = ∠RPA (Alternate interior angles)

but also,

∠RPA = 90° (PQ ⊥ AB) 

⇒ ∠ORP  = 90°

∠ROP + ∠OPA = 180° (Co-interior angles)

⇒∠ROP + 90° = 180°

⇒∠ROP = 90°

Thus, the ΔORP has 2 right angles i.e. ∠ORP  and ∠ROP which is not possible.

Hence, our supposition is wrong. 

∴ PQ passes through the point O.

Answer 2

HELLO
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Let O be the centre of the given circle.
AB is the tangent drawn touching the circle at A.
Draw AC ⊥ AB at point A, such that point C lies on the given circle.
∠OAB = 90° (Radius of the circle is perpendicular to the tangent)
Given ∠CAB  = 90°  
∴ ∠OAB = ∠CAB
This is possible only when centre O lies on the line AC.
Hence, perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.