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 $PA^{2} + PB^{2} =K^{2}$, where k is a constant.

Answer 1

Answer:

x² + y² + z² -2x - 7y + 2z = (K² - 109)/2

Step-by-step explanation:

A (3, 4, 5)

B (-1, 3, -7),

P = (x , y ., z)

PA² = (x - 3)²+ (y - 4)² + (z - 5)²

=> PA² = x² + 9 - 6x + y² + 16 - 8y + z² + 25 - 10z

=> PA² = x² + y² + z² -6x - 8y - 10z + 50

PB² = (x +1)²+ (y - 3)² + (z + 7)²

=> PB² = x² + 1 + 2x + y² + 9 - 6y + z² + 49 + 14z

=> PB² = x² + y² + z² + 2x - 6y + 14z + 59

PA² +  PB² = K²

=> x² + y² + z² -6x - 8y - 10z + 50 + x² + y² + z² + 2x - 6y + 14z + 59 = K²

=> 2x² + 2y² + 2z² -4x - 14y + 4z + 109 = K²

=> x² + y² + z² -2x - 7y + 2z = (K² - 109)/2