Find the joining equation of pair of lines:
Through $(2, - 1)$ and parallel to $2 x^{2} + 3 x y - 9 y^{2} = 0$ .

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Solution

$2 x^{2} + 6 x y - 3 x y - 9 y^{2} = 0$
$2 x (x + 3 y) - 3 y (x + 3 y) = 0$
$(2 x - 3 y) (x + 3 y) = 0$
This is the combined equation of $2 x - 3 y = 0$ and $x + 2 y = 0$
$2 x - 3 y + l_{1} = 0$ $x + 3 y + l_{2} = 0$
$(2, - 1)$ $(2, - 1)$
$4 + 3 + l_{1} = 0$ $2 - 3 + l_{2} = 0$
$l_{1} = 7$ $l_{2} = 1$
$∴$ The equations are
$2 x - 3 y - 7 = 0$ and $x + 3 y + 1 = 0$
And giving equation is
$(2 x - 3 y - 7) (x + 3 y + 1) = 0$
$2 x^{2} - 9 y^{2} + 3 x y - 5 x - 24 y - 7 = 0$ .

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