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 $=2\sqrt{R^2-P^2}$
$=2R$
$=2\sqrt{10}$