Question

If $3x+y+k=0$ is a tangent to the circle $x^2+y^2=10,$ then the values of $k$ are

• A. $\pm 7$
• B. $\pm 5$
• C. $\pm 10$
• D. $\pm 9$

Answer 1

The correct option is C

$\pm 10$

Explanation for correct option:
If the given line is tangent to the circle, then the length of the perpendicular from the center upon the line is equal to the radius of the circle.

$\begin{array}{rcl}\left|\frac{a{x}_1+b{y}_1+c}{\sqrt{a^2+b^2}}\right|& =& r\end{array}$

$\Rightarrow$$\begin{array}{rcl}\left|\frac{3\left(0\right)+1\left(0\right)+k}{\sqrt{3^2+1^2}}\right|& =& \sqrt{10}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\left|\frac{k}{\sqrt{10}}\right|& =& \sqrt{10}\end{array}$

$\Rightarrow$ $\begin{array}{rcl}k& =& \sqrt{100}\end{array}$

$\begin{array}{rcl}\mathbf{\therefore }\mathit{k}& \mathbf{=}& \mathbf{\pm }\mathbf{10}\end{array}$

Hence, The correct option is (C).