Question

The radius of the circle, which is touched by the line $y=x$ and has its centre on the positive direction of x-axis and also cuts-off a chord of length $2$ units along the line $\sqrt{3}y-x=0$, is

• A. $\sqrt{5}$
• B. $\sqrt{3}$
• C. $\sqrt{2}$
• D. $1$

Answer 1

Answer: (C)

$\sqrt{2}$

Since the required circle has its centre on x-axis. So, let its coordinates be $(a,0)$. Also circle touches $y=x$
$\therefore$ Radius$=$ length of $\perp$ from $(a,0)$ to
$y=x$ is $a/\sqrt{2}$
also the circle cuts off a chord of length $2$ units along $x-\sqrt{3}y=0$
$\therefore {CN}^2={MN}^2+{CM}^2$
or ${\left(\frac{a}{\sqrt{2}}\right)}^2=1^2+{\left[\frac{a-\sqrt{3}\times 0}{1^2+{\left(\sqrt{3}\right)}^2}\right]}^2[\because {CN}^2={MN}^2+{CM}^2]$
Radius $=\frac{a}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$