Question

How do I solve for to find the equation of a parabola?

Answer 1

The initial equation is:

$(y-3)^2=(x+2)^2+(y-5)^2$

We can expand the equation to solve it so:

$y^2-9=x^2+4x+4+y^2-25$

and we can simplyfy it:

$\begin{gathered} y^2-y^2=x^2+4x+4-25+9 \\ 0=x^2+4x+12 \end{gathered}$

and we solve the equation so:

$x=\frac{-4\pm\sqrt[]{16-(4)(1)(12)}}{2}$

and we solve:

$\begin{gathered} x=\frac{-4\pm\sqrt[]{-32}}{2} \\ x=\frac{-4\pm4\sqrt[]{2}}{2} \\ x_1=-2+2\sqrt[]{2} \\ x_2=-2-2\sqrt[]{2} \end{gathered}$