How to change $ Cx^2 + Dy^2 + Ex + Fy + G = 0$ to$ (x-h)^2/a^2 ± (y-k)^2/b^2=1 $ using only the variables C, D, E, F, and G

Question

Or, state the terms a,b,h,and k in terms of C, D, E, F, and/or G

$Cx^2 + Dy^2 + Ex + Fy + G = 0$

$(x-h)^2/a^2 ± (y-k)^2/b^2=1$

Answer 1

$$\begin{array}{rcl} \\ Cx^2+Dy^2+Ex+Fy+G&=&0 \\ C(x^2+ \frac EC x)+D(y^2+\frac FD y)&=&-G \\ C(x+\frac E{2C})^2-\frac{E^2}{4C}+D(y+\frac F{2D})^2-\frac{F^2}{4D}&=&-G \\ C(x+\frac E{2C})^2+D(y+\frac F{2D})^2&=&-G+\frac{E^2}{4C}+\frac{F^2}{4D} \end{array}$$ So $$\begin{array}{cr} \\ h=-\frac E{2C} \\a^2= \frac{-G+\frac{E^2}{4C}+\frac{F^2}{4D}}{C} \\ k=-\frac F{2D} \\b^2= \frac{-G+\frac{E^2}{4C}+\frac{F^2}{4D}}{D} \end{array}$$