The lines joining the origin to the point of intersection of $3 x^{2} + m x y - 4 x + 1 = 0$ and $2 x + y - 1 = 0$ are at right angles. Then which of the following is/are possible value/s of $m ?$

- 4

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4

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7

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3

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Solution

The correct options are
A $- 4$
B $4$
C $7$
D $3$
By application of the method of homogenization we get
$3 x^{2} + m x y - 4 x (2 x + y) + 1 (2 x + y)^{2}$
$= 3 x^{2} + m x y - 8 x^{2} - 4 x y + 4 x^{2} + y^{2} + 4 x y$
$= - x^{2} + m x y + y^{2}$
$= 0$
Hence the equations of the lines is given by $x^{2} - m x y - y^{2} = 0$
Hence the lines are perpendicular for all values of $m .$

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