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If tangents to $y^2=4ax$ from P make angles $\theta_1$ and $\theta_2$ with axis and $\theta_1 + \theta_2 = \dfrac\pi4$ then what is locus of $P$?

The answer given is $x-y-1=0$. I tried solving this by using equation for pair of tangents $SS_1=T^2$ which gives $x^2 + hy^2 + x(k^2-2h) - kxy - khy = 0$. I'm stuck over here, what should I be doing next? I already know how to do this problem using the method of expressing tangents in slope form, but I'd like to find out how to do it this way as well. How should I do this? I am aware that this is similar to another question but that question uses polar equation for chord, I would like to know how to do it using combined equation for pair of tangents.

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