A circle having centre $(0, k)$ with $k > 6$ is tangent to the lines pair $x^{2} - y^{2} = 0$ and $y = 6$. The radius of the circle is/ are

The equation x squared minus y squared equals 0 represents a pair of straight lines, y equals x and y equals negative x, intersecting at the origin with angles of 45 degrees to the axes. Because the circle is tangent to both of these lines and the horizontal line y equals 6, its center at 0 comma k lies at an equal distance from all three. The distance from the center to y equals 6 is k minus 6, and using the point-to-line distance formula for y equals x gives the radius as k divided by the square root of 2. Equating these yields k divided by the square root of 2 equals k minus 6, which solves to give a radius of 6 times the quantity square root of 2 minus 1.

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