<s>[INST] Two tangents to the hyperbola $4x^2-y^2=36$ intersect at the point $(0,4)$. Find the coordinates for the points on the hyperbola for this to occur. The question is two tangents to the hyperbola $4x^2-y^2=36$ intersect at the point $(0,4)$. Find the coordinates for the points on the hyperbola for this to occur.

My attempt:
Consider the line $y=mx+c$ as it passes through $(0,4)$ then $c=4$ --> Condition 1
Plug in $y=mx+c$
$$ 4x^2-(mx+c)^2=36$$
$$4x^2-36=m^2x^2+2mcx+c^2$$
$$(m^2-4)x^2+(2mc)x^2+c^2+36=0$$
For tangency $b^2-4ac=0$
Hence
$$ 4m^2c^2-4(c^2+36)(m^2-4)=0$$
$$ 16c^2-144m^2+576=0$$
But $c=4$ due to the line condition hence
$$ m = \pm \frac{2\sqrt{13}}{3}$$
So the lines are 
$y= \frac{2\sqrt{13}}{3}x + 4$ and $y= -\frac{2\sqrt{13}}{3}x + 4$
How to find the coordinates for the points on the hyperbola for this to occur tho?
 [/INST] You already have
$$(m^2-4)x^2+(2mc)x^2+c^2+36=0$$
Solving this gives
$$x=\frac{-mc}{m^2-4}=\frac{-4m}{m^2-4}$$
So, for $m=\pm\frac{2\sqrt{13}}{3}$, the coordinates we want are
$$\color{red}{\left(\mp\frac{3}{2}\sqrt{13},-9\right)}$$
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