Question

Find the equation of the set of points such that the sum of the square of its distance from the points (3,4,5) and (-1, 3, -7) is a constant.

Answer 1

Let (a,b,c) be a set of points whose square of sum of distance from (3,4,5) and (-1,3,-7) is a constant (k)

$(a-3{)}^2+(b-4{)}^2+(c-5{)}^2+(a+1{)}^2+(b-3{)}^2+(c+7{)}^2=k$

$a^2+b^2+c^2-6a-8b-10c+9+16+25+a^2+b^2+c^2+2a-6b+140+1+9+44=k$

$2[a^2+b^2+c^2]-4a-14b+2c+109=k$

$a^2+b^2+c^2-2a-7b+c=\frac{k-109}{2}$

$a^2+b^2+c^2-2a-7b+c=\frac{k-109}{2}$ _______ (1)

Hence equation (1) is the required equation.