Let C be the curve $||z+3\sqrt 2|-|z-3\sqrt 2||=2\sqrt 2$. If locus of $z$ satisfying $|\text{arg} (z-1)|=\tan^{-1} (4)$ meets the curve C at points A and B, then find area of triangle ABP, where P is the point $e^{i2\pi}$
The given curve is a hyperbola with focus $(\pm 3\sqrt 2,0)$
$$\frac{x^2}{2}-\frac{y^2}{16}=1$$
And for $z$
$$\frac{y}{x-1}=4$$
But the locus of $z$ is tangential to the given curve.
I think I am making a computation error or something is wrong with the signs, but I am not able to pinpoint it. Please let me know my mistake