Chord AB of the circle $x^{2} + y^{2} = 100$ passes through the point $(7, 1)$ and subtends an angle 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:

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$\frac{7}{12}$

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answer is A.

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Detailed Solution

Equation of line $- 1 = m (x - 7)$
$m x - y + 1 - 7 m = 0$
Perpendicular distance from $(0, 0) = \frac{r}{2}$
$p = \frac{r}{2}$ or ${(7 m - 1)}^{2} = 25 (1 + m^{2})$
$49 m^{2} - 14 m - 24 = 0$
$\Rightarrow m_{1} m_{2} = - 1$

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