Question

State true or false.

If the line $y=\sqrt{3}x+k$ touches the circle $x^2+y^2=16$, then the value of k is $\pm 8$.

• A. True
• B. False

Answer 1

The correct option is A True

Here the circle has centre at $(0,0)$ and radius 4.

The line touches the circle at a point. So, the distance between this point and the centre is equal to the radius of the circle.

The distance is same as the perpendicular distance from the point $(x_1,y_1)$ to the line $ax+by+c=0$ is given by :

$\frac{|a{x}_1+b{y}_1+c|}{\sqrt{a^2+b^2}}$.

$4=\frac{|0+0+k|}{\sqrt{3+1}}$

$|k|=8$

$k=\pm 8$