The equation of tangent to a circle(in slope form) is given by : $y=mx+a\sqrt{1+m^{2}}$
Tangent(s) pass through the point $\left ( -1,7 \right )$
So, putting the values in tangent equation, we get $7=m\left ( -1 \right )+5\sqrt{1+m^{2}}$
$\Rightarrow$ $m+7=5\sqrt{1+m^{2}}$
$\Rightarrow$ $24m^{2}-14m-24=0$
Let the roots of this equation be $m_{1},m_{2}$.
So, from the quadratic equation, we see that $m_{1}\times m_{2}=-1$ (product of roots is $\frac{c}{a}$)
$\Rightarrow$ Tangents are perpendicular to each other
Option C is correct.