Question

General second degree equation in X and y is ${ax}^2+2hxy+{by}^2+2gx+2fy+c=0$, Where a,h,b,g,f and c are constats.
Prove that condition for it to be a circle is: $a=b$ and $h=0$

Answer 1

General second-degree equation in $X$ and $Y$ is

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$, Where $a,h,b,g,t$ and $c$ are constants. If a $a=b(\ne 0)$ and $h=0$, then the alone equation

becomes: $a{x}^2+a{y}^2+2gx+2fy=0$

$=x^2+y^2+2gx+2fy=0+c$

$\Rightarrow x^2+y^2+2.\frac{g}{a}x+2.\frac{f}{a}y+\frac{c}{a}=0$ (since $a\ne 0$ )

$\Rightarrow x^2+2.x.\frac{g}{a}+\frac{g^2}{a^2}+y^2+2.y.\frac{f}{a}+\frac{f^2}{a^2}=\frac{g^2}{a^2}+\frac{f^2}{a^2}-\frac{c}{a}$

$\Rightarrow (x+\frac{g}{a}{)}^2+(y+\frac{f}{a}{)}^2=(\frac{1}{a}\sqrt{g^2+f^2-Ca}{)}^2$

Which represents the equation of a circle having centre at $(\frac{-g}{a},\frac{-f}{a})$ and radius $=\frac{1}{a}\sqrt{g^2+f^2-Ca}$

$\therefore$ the general second degree equation in $X$ and $Y$

represents a circle if coefficient of $x^2(i.e,a)$=coefficient of $y^2(i.e.b)$ and coefficient of $xy(i,e.h)=0$