Question

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Question : prove that the perpendicular at the point of contact to the tangent to a circle through the centre.

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Answer 1

Answer:

∠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.

Answer 2

Solution:

Draw a circle with centre O, draw a tangent PR touching circle at P.

Draw QP perpendicular to RP at a point P, QP lies in the circle.

Now,

∠OPR = 90º

Also, ∠QPR = 90º

Therefore,

∠OPR = ∠QPR

This is possible only when O lies on QP.

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

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