Question

If the line $y=\sqrt{2}x+k$ touches the circle $x^2+y^2=16,$ then find the value of $k$

Answer 1

The given circle is $x^2+y^2=16$ ..........$\left(1\right)$

Its centre is $C(0,0)$ and its radius is $4$

The given line is $\sqrt{2}x-y+k=0$ ..........$\left(2\right)$

The line $\left(2\right)$ will touch the circle $\left(1\right)$ if the perpendicular distance from $C(0,0)$ to the line $\left(2\right)=$radius of circle.

$\Rightarrow \frac{|\sqrt{2}\times 0-0+k|}{\sqrt{{\left(\sqrt{2}\right)}^2+1^2}}=4$

$\Rightarrow \frac{\left|k\right|}{\sqrt{3}}=4$

$\Rightarrow k=\pm 4\sqrt{3}$