The equation of the plane whose intercepts on $x, y, z$ axes are $1, 2, 4$ respectively is

The plane $\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1$ cuts the axes at the points $A, B, C$. Find the area of triangle $ABC$.

If $M$ is the foot of the perpendicular drawn from $P(1, 2, -1)$ to the plane passing through the point $A(3, -2, 1)$ and perpendicular to the vector $\vec{n} = 4\hat{i} + 7\hat{j} - 4\hat{k}$,then the length of $PM$,in proper units,is

The equation of the plane passing through the points $(2, 3, 1)$ and $(4, -5, 3)$ and parallel to the $X$-axis is:

$A$ variable plane at a constant distance $p$ from the origin meets the coordinate axes at points $A, B, C$. Through these points,planes are drawn parallel to the coordinate planes. Find the locus of their point of intersection.

If $(3,4,-7)$ is the foot of the perpendicular drawn from the point $(-2,3,6)$ to the plane $\pi$,then the sum of the intercepts made by the plane $\pi$ on the $X$ and $Y$-axes is