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(B) Let the coordinates of point $P$ be $(x, y, z)$.Given $PA^2 - PB^2 = k$.Using the distance formula,$PA^2 = (x - 2)^2 + (y - 3)^2 + (z - 4)^2$ and

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$A$ plane

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(B) Let the coordinates of point $P$ be $(x, y, z)$.Given $PA^2 - PB^2 = k$.Using the distance formula,$PA^2 = (x - 2)^2 + (y - 3)^2 + (z - 4)^2$ and $PB^2 = (x + 2)^2 + (y - 5)^2 + (z + 4)^2$.Substituting these into the equation:$[(x - 2)^2 + (y - 3)^2 + (z - 4)^2] - [(x + 2)^2 + (y - 5)^2 + (z + 4)^2] = k$Expanding the squares:$(x^2 - 4x + 4 + y^2 - 6y + 9 + z^2 - 8z + 16) - (x^2 + 4x + 4 + y^2 - 10y + 25 + z^2 + 8z + 16) = k$Simplifying the expression:$(x^2 + y^2 + z^2 - 4x - 6y - 8z + 29) - (x^2 + y^2 + z^2 + 4x - 10y + 8z + 45) = k$$-8x + 4y - 16z - 16 = k$This is a linear equation in $x, y, z$ of the form $ax + by + cz + d = 0$,which represents a plane.

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:

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