Explanation

(c)
The intersection point of line $y = 7x - 25$ and circle $x^{2} + y^{2} = 25$ is $x^{2} + (7x - 25)^{2} = 25$
$\Rightarrow \ \ \ \ 50x^{2} - 350x + 600 = 0$
$\Rightarrow \ \ \ (x - 3)(x - 4) = 0$
$\Rightarrow \ \ x = 3,\ \ x = 4\ \ \Rightarrow y = - 4,\ 3$
$\therefore$ Coordinates of $A(3,\ - 4)$ and $B(4,\ 3)$
$\therefore$ Distance between $A$ and $B = \sqrt{(4 - 3)^{2} + (3 + 4)^{2}}$
$= 5\sqrt{2}$
Alternate Required distance$= 2\sqrt{\frac{a^{2}\left( 1 + m^{2} \right) - c^{2}}{1 + m^{2}}}$
$= 2\sqrt{\frac{25(1 + 49) - 625}{1 + 49}} = 5\sqrt{2}$