Q. A line meets the coordinates axes in $A$ and$B$. A circle is circumscribed about the$\mathrm{\Delta}OAB$. The distances from the points $A$ and $B$ of the side $AB$ to the tangent at $O$ are equal to $m$ and $n$ respectively. Then, the diameter of the circle is
A) $m(m + n)$
B) $n(m + n)$
C) $m - n$
D) None of these
Explanation
(d)
Let the equation of circle be
$x^{2} + y^{2} + 2gx + 2fgy = 0$
(passing through origin)
$Radius = \sqrt{g^{2} + f^{2}}$
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Now, equation of tangents at $O(0,\ 0)$ is
$x(0) + y(0) + g(x) + f(y) = 0$
$\Rightarrow gx + fy = 0$
Distance from $A(2g,\ 0) = \frac{2g^{2}}{\sqrt{g^{2} + f^{2}}} = m$
and distance from $B(0,\ 2f) = \frac{2f^{2}}{\sqrt{g^{2} + f^{2}}} = n$
$\Rightarrow \frac{2r^{2}}{r} = m + n\ \Rightarrow 2r = m + n$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.