Distance Between Two Points

If $A$ and $B$ be the points $(3,4,5)$ and $(-1,3,-7),$ respectively, find the equation of the set of points $P$ such that $P{A}^2+P{B}^2=k^2,$ where $k$ is a constant

Find the coordinates of a point on $y-$axis which is at a distance of $5\sqrt{2}$ from the point $P(3,-2,5)$.

Find the lengths of the medians of the triangle with vertices $A(0,0,6), B(0,4,0)$ and $(6,0,0)$

Verify the following:

$(0,7,-10), (1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle.

Find the equation of the set of points $\mathrm{P}$, the sum of whose distances from $A(4,0,0)$ and $\mathrm{B}(-4,0,0)$ is equal to $10$

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

$(-1,2,1), (1,-2,5), (4,-7,8) \mathrm{and} (2,-3,4)$ are the vertices of a parallelogram.

Verify the following: 

$(0,7,10)$, $(-1,6,6) \mathrm{and} (-4,9,6)$ are the vertices of a right-angled triangle.

Show that the points $(-2, 3, 5), (1, 2, 3)$ and$(7, 0, -1)$are collinear.

If the distance between the points $(2, -1, 3)$ and $(-2, 1, 3)$ is $k\sqrt{5}$ then find the value of $k$.

Find the value of $K$, if the distance between the pair of points $(-1,3,-4)$ and $(1,-3,4)$ is equal to $2\sqrt{K}$.

Find the value of $K^2$, if the distance between the pair of points $(-3,7,2)$ and $(2,4,-1)$ is $K$.

Find the square of the distance between the following pair of points:$(2, 3, 5)$ and $(4, 3, 1)$