Find whether the straight line x - y - 2 - -2 touches the circle x2 - y2 - 2x - 2y - 1 - 0-

Find whether ...

Question

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

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Solution

The circle $x^{2} + y^{2} - 2 x - 2 y + 1 = 0$ 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
$∣ ∣ ∣ ∣ \frac{1 + 1 - 2 - \sqrt{2}}{\sqrt{1^{2} + 1^{2}}} ∣ ∣ ∣ ∣ = 1$ which is equal to the radius of circle.

$∴$ The given line is tangent to the circle.

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