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The equation of the pair of lines passing through the origin whose sum and product of slopes are respectively the arithmetic mean and geometric mean of 4 and 9 is

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$12 x^{2} - 13 xy + 2 y^{2} = 0$

$12 x^{2} - 15 xy + 2 y^{2} = 0$

$12 x^{2} - 15 xy - 2 y^{2} = 0$

$12 x^{2} + 15 xy - 2 y^{2} = 0$

answer is A.

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Detailed Solution

$y^{2} - (m_{1} + m_{2}) xy + m_{1} m_{2} x^{2} = 0$

arthemetic mean :

$\Rightarrow \frac{4 + 9}{2} = \frac{- 2 h}{b} \Rightarrow - \frac{2 h}{b} = \frac{13}{2}$

Geometric mean $= \sqrt{4 (9)} = 6 \Rightarrow \frac{a}{b} = 6$

$\Rightarrow y^{2} - \frac{13}{2} xy + 6 x^{2} = 0 \Rightarrow 2 y^{2} - 13 xy + 12 x^{2} = 0$

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