# Locus of points where tangents drawn on hyperbola make angle 45

Find the locus of points where a pair tangents drawn on the hyperbola $$x^2 - y^2 = a^2$$ enclose an angle of $$45$$ degrees. This is what I've done so far.

$$\theta$$ between tangents is $$45$$ so

$$\lvert \frac{m_2-m_1}{1+m_1m_2)}\rvert = tan45 =1$$
$$\lvert m_2-m_1\rvert = \lvert 1+m_1m_2\rvert$$

I think point-slope form of hyperbola tangent will also be helpful here but I'm not sure how to apply it: $$y = mx \pm \sqrt{a^2m^2-b^2}$$

The equation for pair of tangents can be converted into a quadratic in $$m$$ :
$$(x^2-a^2)m^2-2xy \cdot m + y^2+b^2=0$$
Now use Vieta's formulas on the obtained condition for $$m$$, $$(m_1+m_2)^2-4m_1m_2=(m_1-m_2)^2=(1+m_1m_2)^2$$