Question

Question 6 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 PA^2 + PB^2 = k^2, where k is a constant.

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 279

Answer 1

Let point P ≡ (x , y, z )

A/C to question,
(PA)² + (PB)² = K²

use distance formula ,
PA² = (x - 3)² + (y -4)² + (z - 5)²
PB² = (x + 1)² + (y - 3)² + (z + 7)²

Now,
(x - 3)² + (y - 4)² + (z - 5)² + (x + 1)² + (y - 3)² + (z + 7)² = K²
{(x - 3)² + (x + 1)² } + {(y -4)² + (y - 3)²} + {(z - 5)² + (z + 7)²} = K²
{(2x² -4x + 10} + { 2y² - 14y + 25 } + {2z² +4z + 74 } = K²
2(x² + y² + z²) +( -4x - 14y + 4z) + (10 + 25 + 74) = K²
2(x² + y² + z²) -4x - 14y + 4z + 109 - k² = 0
Which is required equation .