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(A) The equation of a plane passing through the line of intersection of two planes $P_1: 2x - y = 0$ and $P_2: y - 3z = 0$ is given by $P_1 + \lambda

Vedclass · Accepted Answer

$28x - 17y + 9z = 0$

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(A) The equation of a plane passing through the line of intersection of two planes $P_1: 2x - y = 0$ and $P_2: y - 3z = 0$ is given by $P_1 + \lambda P_2 = 0$.$(2x - y) + \lambda(y - 3z) = 0$$2x + (\lambda - 1)y - 3\lambda z = 0$  --- $(i)$This plane is perpendicular to the plane $4x + 5y - 3z - 8 = 0$.The normal vectors of these two planes are $\vec{n_1} = 2\hat{i} + (\lambda - 1)\hat{j} - 3\lambda\hat{k}$ and $\vec{n_2} = 4\hat{i} + 5\hat{j} - 3\hat{k}$.Since the planes are perpendicular,their dot product is zero: $\vec{n_1} \cdot \vec{n_2} = 0$.$2(4) + (\lambda - 1)(5) + (-3\lambda)(-3) = 0$$8 + 5\lambda - 5 + 9\lambda = 0$$14\lambda + 3 = 0$$\lambda = -\frac{3}{14}$Substituting $\lambda = -\frac{3}{14}$ into equation $(i)$:$(2x - y) - \frac{3}{14}(y - 3z) = 0$$28x - 14y - 3y + 9z = 0$$28x - 17y + 9z = 0$.

The equation of the plane containing the line of intersection of the planes $2x - y = 0$ and $y - 3z = 0$ and perpendicular to the plane $4x + 5y - 3z - 8 = 0$ is

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