Find the condition that the lines given by $a^{2} x^{2} + b c y^{2} = a (b + c) x y$ may be parallel to each other .

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Solution

We have,
$\begin{matrix} a^{2} x^{2} + b c y^{2} - a b x y - a c x y = 0 \Rightarrow a^{2} x^{2} - a b x y + b c y^{2} - a c x y = 0 \Rightarrow a x (a x - b y) - c y (a x - b y) = 0 \Rightarrow a x - c y = 0, y = \frac{a}{c} x \Rightarrow a x - b y = 0, y = \frac{a}{b} x \end{matrix}$

These $6$ lines are parallel if $\frac{a}{c} = \frac{a}{b}$

Hence, this is the answer.

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