Find the equation of the lines joining the origin to the points of intersection of the curve $2 x^{2} + 3 x y - 4 x + 1 = 0$ and the line $3 x + y = 1$

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

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

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

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Solution

The correct option is A $x^{2} - y^{2} - 5 x y = 0$
Given equations of curve and the line are $2 x^{2} + 3 x y - 4 x + 1 = 0$ and $3 x + y = 1$ .
Homogenising the curve with the line gives
$2 x^{2} + 3 x y - 4 x (3 x + y) + (3 x + y)^{2} = 0$
$\Rightarrow 2 x^{2} + 3 x y - 12 x^{2} - 4 x y + 9 x^{2} + y^{2} + 6 x y = 0$
$\Rightarrow y^{2} - x^{2} + 5 x y = 0$
$∴$ The equation of the lines joining the origin to the points of intersection of the curve and hte line is $x^{2} - 5 x y - y^{2} = 0$
Hence, option A.

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