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10-1.Circle and System of Circles
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If the tangent to the circle ${x^2} + {y^2} = {r^2}$ at the point $(a, b)$ meets the coordinate axes at the point $A$ and $B$, and $O$ is the origin, then the area of the triangle $OAB$ is
A
$\frac{{{r^4}}}{{2ab}}$
B
$\frac{{{r^4}}}{{ab}}$
C
$\frac{{{r^2}}}{{2ab}}$
D
$\frac{{{r^2}}}{{ab}}$
Solution
(a) Obviously $r = \sqrt {{a^2} + {b^2}} $
Equation of $AB$ is $ax + by = {r^2}$ or $\frac{x}{{{r^2}/a}} + \frac{y}{{{r^2}/b}} = 1$
$ \Rightarrow OA = \frac{{{r^2}}}{a}$ and $OB = \frac{{{r^2}}}{b}$
Hence the area is $\frac{1}{2}.\frac{{{r^2}}}{a}.\frac{{{r^2}}}{b} $
$= \frac{1}{2}\frac{{{r^4}}}{{ab}}$.
Standard 11
Mathematics
