The equation of one of the lines represented by the pair of lines $12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2 = 0$ is/are

6 x - 2 y + 1 = 0

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3 x - y + 2 = 0

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2 x - y + 2 = 0

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4 x - 2 y + 1 = 0

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Solution

The correct options are
B $3 x - y + 2 = 0$
D $4 x - 2 y + 1 = 0$
$12 x^{2} - 10 x y + 2 y^{2} = 0$
$\Rightarrow (6 x - 2 y) (2 x - y) = 0$

So, required equations will be of the form
$6 x - 2 y + λ_{1} = 0$ & $2 x - y + λ_{2} = 0$
Now,
$(6 x - 2 y + λ_{1}) (2 x - y + λ_{2}) = 12 x^{2} - 10 x y + 2 y^{2} + 11 x - 5 y + 2$

By comparing coefficients,
$6 λ_{2} + 2 λ_{1} = 11 2 λ_{2} + λ_{1} = 5 \Rightarrow λ_{1} = 4 and λ_{2} = \frac{1}{2} ∴ line equations are 3 x - y + 2 = 0 and 4 x - 2 y + 1 = 0$

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