Explanation

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