Question

A line meets the coordinate axes in $A$ and $B$. A circle is circumscribed about the triangle $OAB$. If the distance from $A$ and $B$ of the tangent to the circle at the origin be $m$ and $n$, then the diameter of the circle is

• A. $m(m+n)$
• B. $m+n$
• C. $n(m+n)$
• D. $m^2+n^2$

Answer 1

Answer: (B)

$m+n$

Let the coordinate of $a$ be $(a,0)$ and of $B$ be $(0,b)$, then $AOB$ being aright angled triangle the center of the circumscribed circle is mid-point $(\frac{a}{2},\frac{b}{2})$ of $AB$ and radius is $OC=\sqrt{\frac{a^2}{4}+\frac{b^2}{4}}$.

$\therefore$ Equation of the circle is $x^2+y^2-ax-by=0$

Equation of the tangent at the origin is $ax+by=0$ ...(1)

Let $AL$ and $BM$ be the perpendicular from $A$ and $B$ on (1)

then $AL=\left|\frac{a^2}{\sqrt{a^2+b^2}}\right|=m$ and $BM=\left|\frac{b^2}{\sqrt{a^2+b^2}}\right|=n$

$\Rightarrow m+n=\sqrt{a^2+b^2}=$ diameter of the circle.