Question

If $y=3x$ is a tangent to a circle with centre $(1,1)$, then the other tangent drawn through $(0,0)$ to the circle is :

• A. $3y=x$
• B. $y=-3x$
• C. $y=2x$
• D. $y=-2x$

Answer 1

The correct option is A $3y=x$

Given, $y=3x$ is a tangent to a circle with centre $(1,1)$

Let other tangent passing through origin is y=mx

radius of circle $=\left|\frac{1-3}{\sqrt{10}}\right|=\frac{2}{\sqrt{10}}$

$\therefore$ Perpendicular distance from the centre $(1,1)$ to $y=mx$ is equal to the radius.

$\therefore \frac{2}{\sqrt{10}}=\left|\frac{1-m}{\sqrt{1+m^2}}\right|$

$4+4m^2=10(m^2-2m+1)$

$6m^2-20m+6=0$

$3m^2-10m+3=0$

$m=\frac{10\pm \sqrt{100-36}}{6}$

$m=\frac{10\pm 8}{6}$

$\Rightarrow m=3,\frac{1}{3}$

So, other equation of tangent is

$y=\frac{x}{3}$