The equation of the plane bisecting the angle between the planes 3x+4y=4 and 6x−2y+3z+5=0 that contains the origin, is

51 x + 18 y + 15 z - 3 = 0

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9 x - 38 y - 15 z + 53 = 0

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9 x - 2 y - 3 z + 13 = 0

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51 x + 8 y + 5 z - 3 = 0

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Solution

The correct option is A $51 x + 18 y + 15 z - 3 = 0$
Equation of given planes are $P_{1} : 3 x + 4 y - 4 = 0, P_{2} : 6 x - 2 y + 3 z + 5 = 0$
By submitting origin, in $P_{1}$ and $P_{2}$ , we have $P_{1} (0, 0) < 0$ and $P_{2} (0, 0) > 0$
$∴$ Equation of plane of angular bisector containing origin is
$\Rightarrow \frac{a x_{1} + b y_{1} + c z_{1} - d_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}} = - \frac{a x_{1} + b y_{1} + c z_{1} - d_{1}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}}$
$\Rightarrow \frac{3 x + 4 y - 4}{\sqrt{9 + 16 + 0}} = - \frac{6 x - 2 y + 3 z + 5}{\sqrt{36 + 4 + 9}}$
$\Rightarrow \frac{3 x + 4 y - 4}{5} = - \frac{6 x - 2 y + 3 z + 5}{7}$
$\Rightarrow 21 x + 28 y - 28 = - (30 x - 10 y + 15 z + 25)$
$\Rightarrow 51 x + 18 y + 15 z - 3 = 0$

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