If the combined equation of the tangents from $(11,3)$ to the circle $x^2+y^2=65$ is $(11x+3y-65)^2+k(x^2+y^2-65)=0$, find $k$

Question

Using

$$y=mx\pm a\sqrt{1+m^2}$$

I found $m$ to be $\frac 74$ and $-\frac{4}{7}$. From here I can find the separate equations of the tangent, combin them, simplify the given equation and compare. Clearly, it’s extremely tedious and can’t be done easily. Can I use the fact that the tangents are mutually perpendicular to easy my calculations?