If the sum of the squares of the distances of the
point (x, y, z) from the points (a, 0, 0) and (-a, 0, 0)
is `2c^(2)`, then which are of the following is
correct?

Options

A. `x^2+a^2=2c^2-y^2-z^2`
B. `x^2+a^2=c^2-y^2-z^2`
C. `x^2-a^2=c^2-y^2-z^2`
D. `x^2+a^2=c^2+y^2+z^2`

Answer

B

Explanation

Let the point be `P(x, y, z)` and two points, (a,0,0) and (-a,0,0) be A and B
As given in the problem,
`PA^2+PB^2=2c^2`
so, `(x-a)^2+(y-0)^2+(z-0)^2`
`+(x-a)^2+(y-0)^2+(z-0)^2=2c^2`
or, `(x-a)^2+y^2+z^2+(x-a)^2+y^2+z^2=2c^2`
`x^2+2ax+a^2+y^2+z^2+x^2-3ax+a^2+y^2+z^2=2c^2`
`=2(x^2+y^2+z^2+a^2)=2c^2`
`=x^2+y^2+z^2+a^2=c^2`
`=x^2+a^2=c^2-y^2-z^2`