Explanation

(c)
Let the slopes of the two tangents to the hyperbola
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ be $cm$ and $\frac{c}{m}.$
The equations of tangents are
$y = cmx + \sqrt{a^{2}c^{2}m^{2} - b^{2}}$ …(i)
And $my - cx = \sqrt{a^{2}c^{2} - b^{2}m^{2}}$ …(ii)
On squaring and subtracting Eq. (ii) from Eq. (i), we get
$(y - cmx)^{2} - (my - cx)^{2} = a^{2}c^{2}m^{2} - b^{2} - a^{2}c^{2} + b^{2}m^{2}$
$\Rightarrow \left( 1 - m^{2} \right)\left( y^{2} - c^{2}x^{2} \right) = - \left( 1 - m^{2} \right)(a^{2}c^{2} + b^{2})$
$\Rightarrow \ \ y^{2} + b^{2} = c^{2}(x^{2} - a^{2})$