Parabolas passing through $4$ given points in an affine plane

Question

I need to find equations for parabolas which are passing through $(0,-1)$,$(-2,0)$,$(0,3)$,$(4,0)$. Help me. I know that parabolas determined by those four points but how to find equations explicitly?

Answer 1

Use the general parabola equation below,

$$ax^2 + 2\sqrt{ab}xy + b y^2 + gx + fy + c = 0$$

Plug in the points (0,-1) and (0,3) and, without losing generality, let $b=1$ for now,

$$1-f+c, \>\>\>9+3f+c = 0$$

which gives $f=-2$ and $c=-3$. Then, plug in the points (-2,0) and (4,0),

$$4a -2g-3=0,\>\>\>16a+4g-3=0$$

which gives $a=\frac 38$ and $g=-\frac 32$.

Thus, the equation for the parabola is,

$$\frac 38 x^2 + \sqrt{\frac 32 }xy + y^2 -\frac 32 x - 2y -3= 0$$

or, with $b=8$,

$$3 x^2 + 4\sqrt{6}xy + 8y^2 -12 x - 16y -24= 0$$

Edit: As pointed out by @Blue below. A second equation can be obtained by reserving the sign of the cross term.