Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.

Given: Points are A(3, 4, -5) and B(-2, 1, 4)


To find: the equation of the set of the points, i.e. locus of points which are equidistant from the given points


Let the required point P(x, y, z)


According to the question:


PA = PB


⇒ PA2 = PB2


Formula used:


The distance between any two points (a, b, c) and (m, n, o) is given by,



Therefore,


The distance between P(x, y, z) and A(3, 4, -5) is PA,




The distance between P(x, y, z) and B(-2, 1, 4) is PB,




As PA2 = PB2


(x – 3)2+ (y – 4)2 + (z + 5)2 = (x + 2)2 + (y – 1)2 + (z – 4)2


⇒ x2+ 9 – 6x + y2 + 16 – 8y + z2 + 25 + 10z = x2+ 4 + 4x + y2 + 1 – 2y + z2 + 16 – 8z


⇒ x2+ 9 – 6x + y2 + 16 – 8y + z2 + 25 + 10z – x2– 4 – 4x – y2 – 1 + 2y – z2 – 16 + 8z = 0


⇒ – 6x – 6y + 18z + 29 = 0


⇒ 6x + 6y – 18z – 29 = 0


Hence locus of point P is 6x + 6y – 18z – 29 = 0


24