The angle between the tangents from (alpha, b | vedclass

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Question

(C) Let $P = (\alpha, \beta)$ be the external point and $C$ be the center of the circle $(0, 0)$. Let the tangents from $P$ touch the circle at $T_1$

Vedclass · Accepted Answer

$2	an^{-1}\left(\frac{a}{\sqrt{\alpha^2 + \beta^2 - a^2}}ight)$

Vedclass · Answer

(C) Let $P = (\alpha, \beta)$ be the external point and $C$ be the center of the circle $(0, 0)$. Let the tangents from $P$ touch the circle at $T_1$ and $T_2$.In the right-angled triangle $	riangle PT_1C$,the angle $\angle CPT_1 = 	heta$ (where $2	heta$ is the total angle between the tangents).We have $CT_1 = a$ (radius) and $PT_1 = \sqrt{OP^2 - CT_1^2} = \sqrt{\alpha^2 + \beta^2 - a^2}$.Thus,$	an 	heta = \frac{CT_1}{PT_1} = \frac{a}{\sqrt{\alpha^2 + \beta^2 - a^2}}$.Therefore,$	heta = 	an^{-1}\left(\frac{a}{\sqrt{\alpha^2 + \beta^2 - a^2}}ight)$.The total angle between the tangents is $2	heta = 2	an^{-1}\left(\frac{a}{\sqrt{\alpha^2 + \beta^2 - a^2}}ight)$.

The angle between the tangents from $(\alpha, \beta)$ to the circle $x^2 + y^2 = a^2$ is

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