Find the equation of a plane which bisects the line segment joining the points $A(2,3,4)$ and $B(4,5,8)$ at right angles.

If the planes $\bar{r} \cdot(2 \hat{i}-\lambda \hat{j}+\hat{k})=3$ and $\bar{r} \cdot(4 \hat{i}-\hat{j}+\mu \hat{k})=5$ are parallel,then $\lambda+\mu=$

Find the equation of the plane passing through the point $(1, 2, -3)$ and parallel to the plane $3x - 5y + 2z = 11$.

The combined equation for a pair of planes is $S \equiv 2 x^2-6 y^2-12 z^2+18 y z+2 z x+x y=0$. If one of the planes is parallel to $x+2 y-2 z=5$, then the acute angle between the planes $S=0$ is

$(1, -2, 1)$ is a point on a plane $\pi$ and $\pi$ is parallel to the plane $x-y-z=0$. If the equation of $\pi$ is $ax+by+cz-2=0$, then $b-2c=$

Let $A (1, 3, 5)$ and $B (-2, 3, -4)$ be two points. If a point $P(x, y, z)$ moves such that $PA^2 - PB^2 = 6c$,find the locus of $P$.