Let \(a\) and \(b\) be non-zero and real numbers. Then, the equation \(\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0\) represents

Let \(a\) and \(b\) be non- zero real numbers.
Therefore, the given equation \(\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0\) implies either
\[
\begin{array}{rc}
& x^2-5 x y+6 y^2=0 \\
\Rightarrow & (x-2 y)(x-3 y)=0 \\
\Rightarrow & x=2 y \text { and } x=3 y
\end{array}
\]
represent two straight lines passing through origin.
Or \(\quad a x^2+b y^2+c=0\)
When \(c=0\) and \(a\) and \(b\) are of same signs, then
\[
\begin{aligned}
& a x^2+b y^2+c=0 \\
\Rightarrow & x=0 \text { and } y=0
\end{aligned}
\]
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 \(\left(a x^2+b y^2+c\right)\left(x^2-5 x y+6 y^2\right)=0\)
may represent two straight lines and a circle.