Q 39 - Calculus

Question

Find the center and radius of the sphere with the given equation:

$x^2+y^2+z^2+3y-5z+3=0$

Step-by-Step Solution

Answer

The center is $\left(0,\frac{-3}{2},\frac{5}{2}\right)$ and radius is $\frac{\sqrt{22}}{2}$

Given information

A sphere with the equation $x^2+y^2+z^2+3y-5z+3=0$

Finding the center and radius

Converting the equation to whole square terms in x, y and z:

$x^2+y^2+z^2+2\times \frac{3}{2}y-2\times \frac{5}{2}\times z+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2+{\left(\frac{5}{2}\right)}^2-{\left(\frac{5}{2}\right)}^2+3=0$

Segregating the terms:

$x^2+{\left(y+\frac{3}{2}\right)}^2+{\left(z-\frac{5}{2}\right)}^2={\left(\frac{\sqrt{22}}{2}\right)}^2$

Comparing with equation of a sphere, the center is $\left(0,\frac{-3}{2},\frac{5}{2}\right)$and radius is $\frac{\sqrt{22}}{2}$.