Mathematics - Pair of Lines Question with Solution | TestHub

Let $a$ and $b$ be non- zero real numbers. Therefore, the given equation $\left(a{x}^2+b{y}^2+c\right)\left(x^2-5xy+6y^2\right)=0$ implies either $$\begin{array}{rc}& x^2-5xy+6y^2=0\\ \Rightarrow & (x-2y)(x-3y)=0\\ \Rightarrow & x=2y\text{ and }x=3y\end{array}$$ represent two straight lines passing through origin. Or $a{x}^2+b{y}^2+c=0$ When $c=0$ and $a$ and $b$ are of same signs, then $$\begin{array}{rl}& a{x}^2+b{y}^2+c=0\\ \Rightarrow & x=0\text{ and }y=0\end{array}$$ which is a point specified as the origin. When $a=b$ and $c$ is of sign opposite to that of $a,a{x}^2+b{y}^2+c=0$ represents a circle. Hence, the given equation $\left(a{x}^2+b{y}^2+c\right)\left(x^2-5xy+6y^2\right)=0$ may represent two straight lines and a circle.