Question

The equation of sphere through the points $(0,0,0),$ $(-a,b,c),(a,-b,c)$ and $(a,b,-c)$; $a,b,c\ne 0$

• A. $\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x}{a}+\frac{y}{b}+\frac{z}{c}$
• B. $\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x}{a}+\frac{y}{b}-\frac{z}{c}$
• C. $\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x}{a}-\frac{y}{b}+\frac{z}{c}$
• D. $\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{-x}{a}+\frac{y}{b}+\frac{z}{c}$

Answer 1

The correct option is A

$\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x}{a}+\frac{y}{b}+\frac{z}{c}$

Equation of sphere passing through $(0,0,0)$ is given by,
$x^2+y^2+z^2+ux+vy+wz=0$
Given it also passes through $(-a,b,c),(a,-b,c)$ and $(a,b,-c)$
$\Rightarrow a^2+b^2+c^2-ua+vb+wc=0....\left(1\right)$
$\Rightarrow a^2+b^2+c^2+ua-vb+wc=0....\left(2\right)$
$\Rightarrow a^2+b^2+c^2+ua+vb-wc=0....\left(3\right)$
Solving $\left(1\right),\left(2\right)$ and $\left(3\right)$, we get

$u=-\frac{a^2+b^2+c^2}{a},v=-\frac{a^2+b^2+c^2}{b},w=-\frac{a^2+b^2+c^2}{c}$
Therefore, equation of required sphere is

$\frac{x^2+y^2+z^2}{a^2+b^2+c^2}=\frac{x}{a}+\frac{y}{b}+\frac{z}{c}$