Question

Question 4 Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 273

Answer 1

concept : if Q is a point in the plane in such a way that it is equidistant from both the given points A and B , then we have to use the relation for equidistant of two points .
AQ = BQ

Let Q ≡ ( x, y, z )
now,
AQ = √{ (x - 1)² + ( y - 2)² + (z - 3)²}
BQ = √{ (x - 3)² + (y - 2)² + ( z + 1)²}

now,
AQ = BQ
take square both sides,
AQ² = BQ²
(x - 1)² + ( y-2)² + (z - 3)² = (x - 3)² + (y - 2) + (z +1)²
(x - 1)² + (z - 3)² = (x - 3)² + (z + 1)²
(x - 1)² - (x - 3)² = (z + 1)² - (z - 3)²

use formula ,
a² - b² = (a - b)(a + b)

(x - 1 -x + 3)(x - 1 + x - 3) = (z + 1 - z + 3)(z + 1 +z -3)
2(2x -4 ) = 4(2z -2)
4(x - 2) = 8(z - 1)
x - 2 = 2z - 2
x - 2z = 0

hence, required equation is , x - 2z = 0