The length of the chord $x + y = 3$ intercepted by the circle $x^{2} + y^{2} - 2x - 2y - 2 = 0$ is
- $\dfrac {7}{2}$
- $\dfrac {3\sqrt {3}}{2}$
- $\sqrt {14}$
- $\dfrac {\sqrt {7}}{2}$
The circle center is (1, 1) and radius is sqrt(1^2 + 1^2 + 2) = 2. The distance from (1, 1) to x + y - 3 = 0 is |1 + 1 - 3| / sqrt(1^2 + 1^2) = 1/sqrt(2). The half-length of the chord is sqrt(r^2 - d^2) = sqrt(4 - 1/2) = sqrt(7/2). The total length is 2 * sqrt(7/2) = sqrt(14).
The center of the circle x squared plus y squared minus 2x minus 2y minus 2 equals 0 is (1, 1) and its radius squared is 1 plus 1 plus 2, or 4. The perpendicular distance from the center to the line x plus y equals 3 is found using the point-to-line distance formula: the absolute value of 1 plus 1 minus 3 divided by the square root of (1 squared plus 1 squared), which is 1 over the square root of 2. Using the half-chord length formula, half the chord length squared equals the radius squared minus the distance squared, so 4 minus 0.5 equals 3.5. The full length of the chord is 2 times the square root of 3.5, which is the square root of 14.