How to express the parabola $x^2+y^2+2xy-2x-1=0$ in form of SP=PM?

Question

How to express the parabola $x^2+y^2+2xy-2x-1=0$ in form of SP=PM ?

i.e. like: $$(x-a)^2+(y-b)^2= \frac{(mx+ny+c)^2}{m^2+n^2}$$

What is the best method for such a conversion of form of equation ?

Answer 1

The equation for the parabola can be written as $$(x+y)^2=2x+1$$

Because of the symmetry between $x,y$ in the squared term, we suspect that this may be a parabola rotated by $45^{\circ}$.

If we rotate this by another $45^{\circ}$ clockwise using $(x,y)\rightarrow \frac 1{\sqrt 2}(x-y, x+y)$, we have $$y=-\sqrt2\left[\left(x-\frac 1{2\sqrt2}\right)^2-\frac 58\right]$$ which is a translated and reflected version of the parabola $y=\sqrt2 x^2$. This parabola can be written as $x^2=4\left(\frac 1{4\sqrt 2}\right)y$, with focus $\left(0,\frac 1{4\sqrt 2}\right)$, directrix $y=-\frac 1{4\sqrt 2}$, vertex $(0,0)$ and axis of symmetry $x=0$.

Original parabola

Working back, we find that the original parabola has axis of symmetry $x+y=\frac 12$ and vertex $(-\frac 38, \frac 78)$ (where axis of symmetry intersects parabola).

The focus lies on the axis of symmetry, $\frac 1{4\sqrt 2}$ away from the vertex, and is given by $$\color{red}{S\left(-\frac14, \frac 34\right)}.$$

The directrix passes through a point on the axis of symmetry, $\frac 1{4\sqrt 2}$ from the vertex, and away from the parabola, and is perpendicular to the axis of symmetry. The equation of the directrix is given by $y=x+\frac 32$ or $$2x-2y+3=0.$$

To confirm and state the parabola in $SP=PM$ form as required, note that a parametric form of the parabola is $$\color{red}{P(2t^2+2t, 1-2t^2)}$$ which is the coordinate of any point $P$ on the parabola.

We can easily compute that $$\color{red}{SP=\frac {8t^2+4t+1}{2\sqrt 2}}$$

Consider the line perpendicular to the directrix passing through $P$ is given by $x+y=2t+1$. The foot of this perpendicular is given by $$\color{red}{M\left(t-\frac 14, t+\frac 54\right)}$$

It can be easily verified that $$\color{red}{PM=\frac {8t^2+4t+1}{2\sqrt 2}}$$ i.e. $\color{red}{SP=PM}$.

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NB - It may be helpful to plot the curves on desmos.