Question

A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances 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. $\left(\frac{1}{2}\right)(m+n)$

Answer 1

The correct option is B $m+n$

Let the coordinates of $A$ be $(a,0)$ and that of $Bbe(0,b)$

(Fig. 16.10), Since $\angle AOB=\pi /2.$, the line $AB$ is a diameter of the circle circumscribing the triangle $OAB$,. its centre is the mid-point of

$AB$, i.e., $(a/2,b/2),$ and its radius is $\left(1/2\right)$

$AB=\left(l/2\right)\sqrt{a^2+b^2}$. Therefore, equation of the circle through

$O,A$ and $B$ is $x^2+y^2-ax-by=0$ and the equation of the tangent at the origin to this circle is $ax+by=0$.

If $AL$ and $BM$ are the perpendiculars from $A$ and $B$ to this tangent. then

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

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