The equation of the plane passing through $(2, 3, 4)$ and parallel to the plane $5x - 6y + 7z = 3$ is:

The Cartesian equation of the plane passing through the point $(1, -2, 3)$ and perpendicular to the vector $-\hat{i} + 2\hat{j} - 3\hat{k}$ is:

$A$ plane at a unit distance from the origin intersects the three axes at points $P, Q,$ and $R$. If the centroid of $\Delta PQR$ is $(x, y, z)$,and it satisfies the equation $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = k$,then find the value of $k$.

The locus represented by $xy + yz = 0$ is

If the plane $7x + 11y + 13z = 3003$ meets the coordinate axes at $A, B, C$,then the centroid of the $\triangle ABC$ is

$A$ bisector of the angle between the normals of the planes $4x + 3y = 5$ and $x + 2y + 2z = 4$ is along the vector