The length of the foot of the perpendicular from the point $P\left(1, \frac{3}{2}, 2\right)$ to the plane $2x - 2y + 4z + 17 = 0$ is

Prove that if a plane has the intercepts $a, b, c$ and is at a distance of $p$ units from the origin,then $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}$.

Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a, b, c$ and $a^{\prime}, b^{\prime}, c^{\prime}$ respectively from the origin,prove that $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$.

$A$ plane passes through the point $A(2, 1, -3).$ If the distance of this plane from the origin is maximum,then its equation is

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

Let two planes be $P_1: 2x - y + z = 2$ and $P_2: x + 2y - z = 3$. Find the equation of the plane passing through the point $(-1, 3, 2)$ and perpendicular to both planes $P_1$ and $P_2$.