If angle between two tangents drawn


If angle between two tangents drawn from a point P  to a circle of radius a and centre 0 is 90°, then OP = a √2.



From point $P$, two tangents are drawn.

Given, $O T=a$

Also, line OP bisects the $\angle R P T$.

$\therefore$ $\angle T P O=\angle R P O=45^{\circ}$

Also, $O T \perp P T$

In right angied $\triangle O T P$. $\sin 45^{\circ}=\frac{O T}{O P}$

$\Rightarrow$ $\frac{1}{\sqrt{2}}=\frac{a}{O P} \Rightarrow O P=a \sqrt{2}$

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