How to get standard form of parabola from oblique parabola??

Question

The given parabloa is $x^2+2xy+y^2+4x-4y-4=0$. How do I convert it to $x^2=4ax$ or $y^2=4ay$ (aka standard form)?

Answer 1

The standard parabola can be written as $L_1^2 =4A L_2$, where $L_1,L_2$ are perpendicular and normalized lines.The length of lataus rectum is $4A$ So here we have $$x^2-+xy+y^2+4x-4y-4=0 \implies (x+y)^2=4(y-x+1). $$ $$\implies\left(\frac{(x+y)}{\sqrt{2}}\right)^2=2\sqrt{2}\frac{(y-x+1)}{\sqrt{2}}$$ The length of the latus rectum is $2\sqrt{2}$ the equation of the axis of the parabola is $x+y=0$ and Tangent at vertex is $y-x+1=0$.