jmc

Question

A tangent from the point (2 1994, 2 1994) to the circle x^2 + y^2 = 1994^2 touches circle at the point (a,b). Compute a + b.

Accepted Answer

Let r = 1994. The line joining the center (0,0) to (a,b) is perpendicular to the line joining (2r,2r) to (a,b). Hence, the product of their slopes is -1. FIGURE This gives us the equation 2r - b2r - a ba = -1.Then b(2r - b) = -a(2r - a), which expands as 2br - b^2 = -2ar + a^2. Then 2ar + 2br = a^2 + b^2 = r^2, so a + b = r^22r = r2 = 997.