Find the equation of the set of points which are equidistant from the points $(1, 2, 3)$ and $(3, 2, -1)$.

The equation of the plane passing through the points $(1, 2, 3)$,$(-1, 4, 2)$ and $(3, 1, 1)$ is

$A$ plane passing through the points $A(2, 3, 5)$ and $B(-3, -5, -7)$ is perpendicular to the plane $x - y + z = 1$. Which of the following points lies on this plane?

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 $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{a^{\prime 2}}+\frac{1}{b^{\prime 2}}+\frac{1}{c^{\prime 2}}$.

The length of the perpendicular drawn from the point $(2, 1, 4)$ to the plane containing the lines $\vec r = (\hat i + \hat j) + \lambda (\hat i + 2\hat j - \hat k)$ and $\vec r = (\hat i + \hat j) + \mu (-\hat i + \hat j - 2\hat k)$ is

The equation of the plane passing through $(-1, 1, 2)$ whose normal makes equal acute angles with the coordinate axes is: