Question

A line meets the coordinate axes in $A$ and $B$. A circle is circumscribed about the $△AOB$. If $m,n$ are the distance of the tangent to the circle at the origin from the points $A$ and $B$, respectively the diameter of the circle is

• A. $m(m+n)$
• B. $m+n$
• C. $n(m+n)$
• D. None of these

Answer 1

Answer: (B)

$m+n$

Let $A\equiv (a,0)$ and $B\equiv (0,b)$

Since $\angle AOB={90}^0$, $\therefore AB$ is the diameter

$\therefore$ Center of the circle is and radius $=\frac{1}{2}\sqrt{a^2+b^2}$.

$\therefore$ Equation of the circle is

${(x-\frac{a}{2})}^2+{(y-\frac{b}{2})}^2=\frac{1}{4}(a^2+b^2)\Rightarrow x^2+y^2-ax-by=0$

Equation of tangent to the circle at $O(0,0)$ is $ax+by=0$ ...(1)

$\therefore m=$ length of $\perp$ from $A(a,0)$ on (1) $=\frac{a^2}{\sqrt{a^2+b^2}}$

and $n=$ length $\perp$ from $(0,b)$ on (1) $=\frac{b^2}{\sqrt{a^2+b^2}}$

$\therefore$ Diameter $\sqrt{a^2+b^2}=m+n$