Q 70. - Calculus

Question

Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b csc \theta$ in cylindrical coordinates.

Step-by-Step Solution

Answer

Conversion of equation is done using $x=r\cos \theta ,y=r\sin \theta ,z=z$.

Step 1: Given Information

Equation of plane in rectangular coordinates is given by $y=b$ ($b$ is constant)

Step 2: Simplification

We know that

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

For the transformation, use $y=r\sin \theta$ in $y=b$

$r\sin \theta =b$

$r=\frac{b}{\sin \theta }$

$\Rightarrow r=b\csc \theta$