Question

A circle touches $y$ -axis at $\left(0 , 3\right)$ and makes an intercept of $2$ units on the positive $x$ -axis. intercept made by the circle on the line $\sqrt{10}x-3y=1$ in units is

• A. $3$
• B. $6$
• C. $2\sqrt{10}$
• D. $\sqrt{10}$

Answer 1

Answer: (C)

$R=$ Radius $=\sqrt{10}$
centre $\left(\sqrt{10} , 3\right)$
$S=$ Equation of circle
$\Rightarrow \left(x - \sqrt{10}\right)^{2}+\left(y - 3\right)^{2}=10$
Intercept made by circle $S=0$ on line
$\sqrt{10}x-3y=1$
Perpendicular distance $P=\left|\frac{10 - 9 - 1}{\sqrt{19}}\right|=0$
Length of intercept $=2\sqrt{R^{2} - P^{2}}$
$=2R$
$=2\sqrt{10}$