please help me solve the question !!

Question

Given two points A (1,2,3), B (7,6,5), let S be the set of all points in (x, y)–plane such that PA is orthogonal to PB, then >>

 

(a) S is empty     (b) S contains exactly one point      (c) S contains exactly two points      (d) S is a line segment    (e) S is a circle

Answer 1

Every point in the xy-plane has z-coordinate equal to zero.

 

P = (x,y,0)

 

Let v be the vector from A to P and w be the vector from B to P.

 

     v = < x-1, y-2, -3 >   w = <x-7, y-6, -5>

 

Since v and w are orthogonal, v•w = 0

 

So, (x-1)(x-7) + (y-2)(y-6) + 15 = 0

 

      x² - 8x + 7 + y² - 8y + 12 + 15 = 0

 

      (x²-8x) + (y²-8y) = -34

 

      (x² - 8x + 16) + (y² - 8y + 16) = -2

 

               (x-4)² + (y-4)² = -2

 

Since the left side is never negative, there are no points, P, in the xy-plane with PA orthogonal to PB.