A and B are two points in a three dimensional space. A point P moves in the space such that
PA = PB. What is the locus of P.
Answers
locus of point P is a plane bisecting AB at right angle as shown in figure.
let A(a1, a2, a3) and B(b1, b2, b3) are located in a three dimensional space.
a point P(x,y) moves in three dimensional space such that PA = PB
so, PA² = PB²
from distance formula,
PA² = (x - a1)² + (y - a2)² + (z - a3)²
PB² = (x - b1)² + (y - b2)² + (z - b3)²
so, (x - a1)² + (y - a2)² + (z - a3)² = (x - b1)² + (y - b2)² + (z - b3)²
⇒(x - a1-x + b1)(x -a1 + x - b1) + (y - a2 - y + b2)(y - a2 + y - b2) + (z - a3 - z + b3)(z - a3 + z - b3) = 0
⇒(b1 -a1)(2x - a1-b1) + (b2 - a2)(2y - a2 - b2) + (b3 - a3)(2z - a3 - b3) = 0
⇒2(b1 - b2)x + 2(b2 - a2)y + 2(b3 - a3)z - [(b1 - a1)(a1 + b1) + (b2 - a2)(a2 + b2) + (b3 - a3)(a3 + b3)] = 0, equation of plane.
hence, locus of point P is a plane bisecting AB at right angle.
also read similar questions : A(1,3) B(2,1) aer point if the equation of the locus of point P such that PA=PB
https://brainly.in/question/11591959
if A(-3,2) and B(1,-4), find the equation of the locus of point P such that 3(PA)=2( PB)
https://brainly.in/question/2524144
