Chord $AB$ of the circle ${x^2} + {y^2} = 100$ passes through the point $\left( {7,1} \right)$ and subtends an angles of ${60^0}$ at the circumference of the circle. If ${m_1}$ and ${m_2}$ are the slopes of two such chords then the value of ${m_1}{m_2}$ is

The angle subtended by a chord at the circumference is half the angle at the center. If it subtends 60 degrees at the circumference, it subtends 120 degrees at the center. The distance from the center (0,0) to the chord is d = R * cos(120/2) = 10 * cos(60) = 5. The line equation passing through (7,1) with slope m is y - 1 = m(x - 7) or mx - y + (1 - 7m) = 0. Distance from origin = |1 - 7m| / sqrt(m^2 + 1) = 5. Squaring: (1 - 7m)^2 = 25(m^2 + 1). 1 - 14m + 49m^2 = 25m^2 + 25. 24m^2 - 14m - 24 = 0. Product of roots m1*m2 = c/a = -24/24 = -1.