If line y cos α = xsin α + acos α be a tangent to circle {x^2} + {y^2} = {a^2} , then

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10-1.Circle and System of Circles

medium

If the line $y$ $\cos \alpha = x\sin \alpha + a\cos \alpha $ be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then

A

${\sin ^2}\alpha = 1$

B

${\cos ^2}\alpha = 1$

C

${\sin ^2}\alpha = {a^2}$

D

${\cos ^2}\alpha = {a^2}$

Solution

(b) The tangent is $y$ $\cos \alpha = x\sin \alpha + a\cos \alpha $

$ \Rightarrow y = x\tan \alpha + a$

It is a tangent to the circle ${x^2} + {y^2} = {a^2}$, if

${a^2} = {a^2}(1 + {\tan ^2}\alpha ) $

$\Rightarrow {\sec ^2}\alpha = 1 $

$\Rightarrow {\cos ^2}\alpha = 1$.

Standard 11

Mathematics

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