The area of the triangle formed by the tangen | vedclass

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Question

(a) Equation of chord of contact $AB $ is

$xh + yk = {a^2}$.....$(i)$

$OM = $length of perpendicular from $O(0, 0)$ on line $(i)$

$ = \frac{{

Vedclass · Accepted Answer

$a{\rm{ }}\frac{{{{({h^2} + {k^2} - {a^2})}^{3/2}}}}{{{h^2} + {k^2}}}$

Vedclass · Answer

(a) Equation of chord of contact $AB $ is

$xh + yk = {a^2}$.....$(i)$

$OM = $length of perpendicular from $O(0, 0)$ on line $(i)$

$ = \frac{{{a^2}}}{{\sqrt {{h^2} + {k^2}} }}$

$	herefore $ $AB = 2AM = 2\sqrt {O{A^2} - O{M^2}}  $

$= \frac{{2a\sqrt {{h^2} + {k^2} - {a^2}} }}{{\sqrt {{h^2} + {k^2}} }}$

Also $PM = $length of perpendicular from $P(h,\;k)$to the line $(i)$ is $\frac{{{h^2} + {k^2} - {a^2}}}{{\sqrt {{h^2} + {k^2}} }}$

Therefore, the required area of triangle $PAB$

$ = \frac{1}{2}.\;AB\;.\;PM $

$= \frac{{a{{({h^2} + {k^2} - {a^2})}^{3/2}}}}{{{h^2} + {k^2}}}$.

The area of the triangle formed by the tangents from the points $(h, k)$ to the circle ${x^2} + {y^2} = {a^2}$ and the line joining their points of contact is

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