The coordinates of the points $A$ and $B$ are $(2, 3, 4)$ and $(-2, 5, -4)$ respectively. If a point $P(x, y, z)$ moves such that $PA^2 - PB^2 = k$,where $k$ is a constant,then the locus of $P$ is:

If a plane passing through the points $(2,3,0), (0,-5,2)$ and $(-2,0,3)$ meets the $X, Y, Z$-axes in $A, B, C$ respectively, then $A=$

If $(x, y, z)$ is an arbitrary point lying on a plane $P$ which passes through the points $(42, 0, 0)$,$(0, 42, 0)$,and $(0, 0, 42)$,then the value of the expression $3 + \frac{x-11}{(y-19)^{2}(z-12)^{2}} + \frac{y-19}{(x-11)^{2}(z-12)^{2}} + \frac{z-12}{(x-11)^{2}(y-19)^{2}} - \frac{x+y+z}{14(x-11)(y-19)(z-12)}$ is:

Find the vector equation of the plane passing through the point $2\hat{i} + \hat{j} - 4\hat{k}$ and parallel to the plane $\vec{r} \cdot (4\hat{i} - 12\hat{j} - 3\hat{k}) - 7 = 0$.

If the equation of a plane passing through $A(1, p, 2)$ and $B(3, 2, 4)$ and parallel to the $z$-axis is $3x - 2y - q = 0$, then:

The direction ratios of the normal to the plane passing through the origin and the line of intersection of the planes $x+2y+3z=4$ and $4x+3y+2z=1$ are . . . . . .