2023-02-24

How to decide whether or not the equation below has a circle as its graph If it does, give the center and the radius. If it does not, describe the graph $25{x}^{2}+25{y}^{2}-30x+30y-18=0$?

Aubree Phelps

Equation of the General Second Degree in ${ℝ}^{2}$
$a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ will represent a Circle , if,
$\left(i\right):a=b\ne 0,\left(ii\right):h=o,&,\left(iii\right):{g}^{2}+{f}^{2}-ac>0.$
In the event, its Centre is $\left(-\frac{g}{a},-\frac{f}{a}\right)$ and Radius is
$\frac{\sqrt{{g}^{2}+{f}^{2}-ac}}{|a|}$.
In our Example, $\left(i\right):a=25=b\ne 0,\left(ii\right):h=0,$ and,
$\left(iii\right):g=-15,f=15,c=-18$
$⇒{g}^{2}+{f}^{2}-ac=225+225=900>0$.
So, the eqn. represents a circle having centre $\left(15/25,-15/25\right)$, i.e.,
$\left(\frac{3}{5},-\frac{3}{5}\right)\phantom{\rule{1ex}{0ex}}\text{& radius}\phantom{\rule{1ex}{0ex}}\frac{\sqrt{900}}{|25|}=\frac{30}{25}=\frac{6}{5}$.

Do you have a similar question?