Question

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

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

\[\therefore 4 = \left| \frac{\sqrt{3}\left( 0 \right) - 0 + k}{\sqrt{\left( \sqrt{3} \right)^2 + 1^2}} \right|\]
\[ \Rightarrow \pm 4 = \frac{k}{2}\]
\[ \Rightarrow k = \pm 8\]