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The radius is always perpendicular to …

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The radius is always perpendicular to the radius at the point of contact

Posted by Abhishek Gupta 6 years, 6 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 6 months ago

Given A radius OP of a circle C (O, r) and a line APB, perpendicular to OP.
To Prove AB is a tangent to the circle at the point P.

PROOF Take a point Q, different from P, on the line AB.since radius through the point of contact of tangent is perpendicular to it. Therefore,
{tex} O P \perp A B{/tex}.
{tex}{/tex}We know that among all the line segments joining O to a point on AB, OP is the shortest one. Therefore,
{tex}{/tex} OP < <oq>OQ
{tex}{/tex}{tex}\Rightarrow{/tex} OQ > OP</oq>
{tex}\Rightarrow{/tex}Q <oq>lies outside the circle.
Therefore, every point on AB, other than P, lies outside the circle. This implies that AB meets the circle only at the point P.
Hence, AB is a tangent to the circle at P.</oq>

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