Question

if A and B be the point (3,4,5) and (-1,3,-7) respectively. find the equation of the set of point p such that PA² + PB² = k², where k is a constant.​

Answer 1

Given:-

• A and B be the point (3,4,5) and (-1,3,-7).
• The set of point p such that PA² + PB² = k²

To find:-

• Find the equation ..?

Solutions:-

• The coordinates of point A and B are given as (3, 4, 5) and (-1, 3, -7).

• Let the coordinates of point P be (x, y, z).

On using distance, 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 + 50

PB² = (x + 1)² + (y - 3)² + (z - 7)²

= x² + 2x + y² - 6y + z² + 14z + 59

Now, PA² + PB² = K²

=> (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 = K² - 109 / 2

Hence, the required equation is 2(x² + y² + z² - 2x - 7y + 2z) = K² - 109 / 2.