Equation of a line which is parallel to the line common to the pair of lines given by $6 x^{2} - x y - 12 y^{2} = 0$ and $15 x^{2} + 14 x y - 8 y^{2} = 0$ and the sum of whose intercepts on the axes is $7$ , is

2 x - 3 y = 42

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3 x + 4 y = 12

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5 x - 2 y = 10

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none of these

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Solution

The correct option is B $3 x + 4 y = 12$
$6 x^{2} - x y - 12 y^{2} = 0$
$\Rightarrow (2 x - 3 y) (3 x + 4 y) = 0 . . . (i)$ and
$15 x^{2} + 14 x y - 8 y^{2} = 0 \Rightarrow (5 x - 2 y) (3 x + 4 y) = 0... (i i)$
Equation of the line common to $(i)$ and $(i i)$ is $3 x + 4 y = 0 . . . (i i i)$
Equation of any line parallel to $(i i)$ is
$3 x + 4 y = k$ or $\frac{x}{k / 3} + \frac{y}{k / 4} = 1$
If $\frac{k}{3} + \frac{k}{4} = 7,$ then $k = 12$ and the equation of the
required line is $3 x + 4 y = 12.$

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