Question

Find whether the straight line $x+y=2+\sqrt{2}$ touches the circle $x^2+y^2-2x-2y+1=0$.

Answer 1

The circle $x^2+y^2-2x-2y+1=0$ is centred at $(1,1)$ and has radius equal to $1$.

For line to be tangent, it's perpendicular distance from $(1,1)$ should be equal to the radius of circle.

Perpendicular distance of $x+y=2+\sqrt{2}$ from $(1,1)$ is

$\left|\frac{1+1-2-\sqrt{2}}{\sqrt{1^2+1^2}}\right|=1$ which is equal to the radius of circle.

$\therefore$ The given line is tangent to the circle.