The locus of the mid-points of the chords of the circle $x^{2} + y^{2} + 2x - 2y - 2 = 0$ which make an angle of $90^{\circ}$ at the centre is
- $x^{2} + y^{2} - 2x - 2y = 0$
- $x^{2} + y^{2} - 2x + 2y = 0$
- $x^{2} + y^{2} + 2x - 2y = 0$
- $x^{2} + y^{2} + 2x - 2y - 1 = 0$
Circle: (x+1)^2 + (y-1)^2 = 4. Center (-1, 1), radius 2. Chord makes 90 degrees at center, so distance from center to chord d = r/sqrt(2) = 2/sqrt(2) = sqrt(2). Midpoint (h, k) distance from center is sqrt(2). (h+1)^2 + (k-1)^2 = (sqrt(2))^2 = 2. h^2 + 2h + 1 + k^2 - 2k + 1 = 2. h^2 + k^2 + 2h - 2k = 0.
The given circle equation is x squared plus y squared plus 2x minus 2y minus 2 equals 0, so its centre is (-1, 1) and its radius squared is 1 plus 1 plus 2, which is 4. A chord that subtends a 90 degree angle at the centre has a perpendicular distance d from the centre given by d equals r times cos(45 degrees), which equals 2 times (1 divided by the square root of 2), or the square root of 2. The locus of the midpoint (h, k) of such chords is a concentric circle, so we use the distance formula to equate the squared distance from (-1, 1) to (h, k) with the squared distance 2. This yields h plus 1 squared plus k minus 1 squared equals 2, which expands to h squared plus k squared plus 2h minus 2y equals 0. Replacing h with x and k with y gives the final equation x squared plus y squared plus 2x minus 2y equals 0.