Question

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 PA²+PB²= k², where k is a constant.

Answer 1

The coordinates of points A and B are given as (3, 4, 5) and (-1, 3, -7) respectively.
Let the coordinates of point P be (x, y, z).
On using distance formula, we obtain
PA² = (x-3)²+(y-4)²+(z−5)²
= x²+9-6x+ y²+16-8y+z²+25-10z
= x²-6x +y²-8y ^{+ z2-10z} 
PB2 = (x+1)²+(y-3)²+(z+7)²
= x²+2x+ y²−6y+z²+14z+59
Now, if PA²+PB²=k², then
(x²-6x+ y²-8y+z²-10z+50)+(x²+2x+ y² −6y+z²+14z+59) = k² 
⇒2x²+2y²+2z²-4x-14y+4z+109= k²
⇒2(x²+ y²+z²-2x-7y+2z)=k²-109
⇒x² + y²+ z² -2x -7y +2z =\(\frac{k^2-109}{2}\)
Thus, the required equation is x²+y²+z²-2x -7y +2z = \(\frac{k^2-109}{2}\)