Question

Describe the graphs of the equations and provide alternative equations in the specified coordinate systems

Change$\rho =2$ to the rectangular and cylindrical systems

Short answer

The equation in rectangular coordinates is$x^2+y^2+z^2=4$

The equaion in cylindrical coordinates is$r^2+z^2=4$

Step-by-step solution

Step 1:Given information

The given expression is$\rho =2$

Step 2:Simplification

To convert $\rho =2$into rectangular coordinates

In rectangular coordinates we have

$x^2+y^2+z^2={\rho }^2$

Since $\rho =2$

$\Rightarrow x^2+y^2+z^2=4$

So, it represents a sphere with center at origin

Step 3:Simplification

To convert $\rho =2$into cylindrical coordinates we have

$x=r\cos \theta ,y=r\sin \theta ,z=z$

substituting in the above equation

width="198" height="22" role="math">⇒r2cos2θ+r2sin2θ+z2=4

width="195" height="22" role="math">⇒r2cos2θ+sin2θ+z2=4

$\Rightarrow r^2+z^2=4$

Therefore,the equaion in cylindrical coordinates is$r^2+z^2=4$