Question

If the line y = $\sqrt{3}$x + k touches the circle x² + y² = 16, then find the value of k. [NCERT EXEMPLAR]

Answer 1

The centre and the radius of the circle x² + y² = 16 are (0, 0) and 4
Now, the perpendicular distance from the centre of the circle to the tangent y = $\sqrt{3}$x + k is equal to the radius of the circle
$$\begin{gathered} \therefore 4=\left|\frac{\sqrt{3}\left(0\right)-0+k}{\sqrt{{\left(\sqrt{3}\right)}^2+1^2}}\right| \\ \Rightarrow \pm 4=\frac{k}{2} \\ \Rightarrow k=\pm 8 \end{gathered}$$