If A and B be the points (3, 4, 5) and (-1, 3, -7) respectively, find

1.	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 constant.
Answer» Given points are A (3, 4, 5) and B (-1, 3, -7). Let P = (x, y, z) Given: PA² + PB² = k² ⇒ (x - 3)² + (y - 4)² + (z - 5)² +(x + 1)² + (y – 3)² + (z + 7)² =k² ⇒ 2x¹ – 6x + 9 + y² -8y + 16 + z² -10z + 25 + x² + 2x + 1 + y² – 6y + 9 + z² +14z + 49 = k² ⇒ 2x² + 2y² + 2z² – 4x – 14y + 4z +109 = k² ∴ Required equations of the set of points P is, 2x² + 2y² +2z² – 4x – 14y + 4z + 109 – k² =0

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 constant.

Answer»

Given points are A (3, 4, 5) and B (-1, 3, -7).

Let P = (x, y, z)

Given: PA² + PB² = k²

⇒ (x - 3)² + (y - 4)² + (z - 5)² +(x + 1)² + (y – 3)² + (z + 7)² =k²

⇒ 2x¹ – 6x + 9 + y² -8y + 16 + z² -10z + 25 + x² + 2x + 1 + y² – 6y + 9 + z² +14z + 49 = k²

⇒ 2x² + 2y² + 2z² – 4x – 14y + 4z +109 = k²

∴ Required equations of the set of points P is,

2x² + 2y² +2z² – 4x – 14y + 4z + 109 – k² =0