Question

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

• A. m(m + n)
• B. m + n
• C. n(m + n)
• D.

Answer 1

The correct option is B m + n


Let the coordinates of A be (a, 0) and of B be (0, b)
Then AOB being a right angled triangle AB is a diameter of the circle, so equation of the circle is
(x – a) (x – 0) + (y – b) (y + b) (y – 0) = 0
$\Rightarrow x^2+y^2-ax-by=0$
Equation of the tangent at the origin is $ax+by=0\to \left(1\right)$
Then 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