Joint equation of a pair of lines passing through the point of intersection of the lines $x^{2} + x y - 2 y^{2} - 4 x + 7 y - 5 = 0$ and perpendicular to these lines is

2 x^{2} - x y + y^{2} - 4 x + 7 y - 5 = 0

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4 x^{2} - 7 x y - 2 y^{2} + 6 x + 15 y - 18 = 0

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2 x^{2} + x y - y^{2} + 2 x + y + 5 = 0

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none of these

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Solution

The correct option is D none of these
$x^{2} + x y - 2 y^{2} - 4 x + 7 y - 5 = 0 \Rightarrow (x - y + 1) (x + 2 y - 5) = 0$
Intersection point is $(1, 2)$
Equation of line passing through $(1, 2)$ and perpendicular to $x - y + 1 = 0$ is
$x + y - 3 = 0$
And equation of line passing through $(1, 2)$ and perpendicular to $x + 2 y - 5 = 0$ is
$2 x - y = 0$
Hence their joint equation is
$(x + y - 3) (2 x - y) = 0$

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