Guadalupe Cooke

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

Beginner2023-02-25Added 7 answers

Equation of the General Second Degree in $\mathbb{R}}^{2$

$a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ 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 $(-\frac{g}{a},-\frac{f}{a})$ and Radius is

$\frac{\sqrt{{g}^{2}+{f}^{2}-ac}}{\left|a\right|}$.

In our Example, $\left(i\right):a=25=b\ne 0,\left(ii\right):h=0,$ and,

$\left(iii\right):g=-15,f=15,c=-18$

$\Rightarrow {g}^{2}+{f}^{2}-ac=225+225=900>0$.

So, the eqn. represents a circle having centre $(15/25,-15/25)$, i.e.,

$(\frac{3}{5},-\frac{3}{5})\phantom{\rule{1ex}{0ex}}\text{\& radius}\phantom{\rule{1ex}{0ex}}\frac{\sqrt{900}}{\left|25\right|}=\frac{30}{25}=\frac{6}{5}$.

$a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c=0$ 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 $(-\frac{g}{a},-\frac{f}{a})$ and Radius is

$\frac{\sqrt{{g}^{2}+{f}^{2}-ac}}{\left|a\right|}$.

In our Example, $\left(i\right):a=25=b\ne 0,\left(ii\right):h=0,$ and,

$\left(iii\right):g=-15,f=15,c=-18$

$\Rightarrow {g}^{2}+{f}^{2}-ac=225+225=900>0$.

So, the eqn. represents a circle having centre $(15/25,-15/25)$, i.e.,

$(\frac{3}{5},-\frac{3}{5})\phantom{\rule{1ex}{0ex}}\text{\& radius}\phantom{\rule{1ex}{0ex}}\frac{\sqrt{900}}{\left|25\right|}=\frac{30}{25}=\frac{6}{5}$.