Question

The equation of a sphere having centre $(-1,2,-3)$ and radius $3$ units is

• A. $x^2+y^2+z^2–2x–4y–6z=0$
• B. $x^2+y^2+z^2+2x–4y+6z+5=0$
• C. $x^2+y^2+z^2–2x–4y–6z–5=0$
• D. None of these

Answer 1

The correct option is B

$x^2+y^2+z^2+2x–4y+6z+5=0$

Explanation for the correct option:

Finding the equation of a sphere:

Equation of sphere with center $(a,b,c)$and radius $r$ is given by

$\begin{array}{rcl}{(x-a)}^2+{(y-b)}^2+{(z-c)}^2& =& r^2\end{array}$

Here, the equation of sphere with center $(-1,2,-3)$and radius $3$ is given by

$\begin{array}{rcl}{(x+1)}^2+{(y–2)}^2+{(z+3)}^2& =& 3^2\\ x^2+1+2x+y^2+4-4y+z^2+9+6z& =& 9\left[\text{Using}{\left(a+b\right)}^2\text{and}{\left(a-b\right)}^2\text{formula}\right]\\ x^2+2x+y^2-4y+z^2+6z+1+4+9& =& 9\\ x^2+y^2+z^2+2x–4y+6z+5& =& 0\end{array}$

Hence, option (B) is the correct answer.