If the equation of one tangent to the circle with centre at $(2,-1)$ from the origin is $3 x+y=0$, then the equation of the other tangent through the origin is

• A

  $3 x-y=0$
• B

  $x+3 y=0$
• Correct answer
  C

  $x-3 y=0$
• D

  $x+2 y=0$

Solution

$$ \begin{aligned} &\because \mathrm{CA}=\mathrm{CB} \\ &\Rightarrow \frac{5}{\sqrt{10}}=\left|\frac{2 \mathrm{~m}+1}{\sqrt{1+\mathrm{m}^2}}\right| \end{aligned} $$ squaring $$ \frac{25}{10}=\frac{4 m^2+4 m+1}{1+m^2} $$ $$ \Rightarrow 5+5 m^2=8 m^2+8 m+2 \Rightarrow 3 m^2+8 m-3=0 \Rightarrow m=\frac{1}{3},-3 $$ $\therefore$ Equation of tangent $\mathrm{OB}$ is $y=\frac{x}{3}$

Hence, the correct answer is option: 'C'.