Q 69. - Calculus

Question

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

Step-by-Step Solution

Answer

Use the transformation $x=r\cos \theta ,y=r\sin \theta ,z=z$ to prove the mentioned relation.

Step 1: Given Information

Equation of plane in rectangular coordinates is given as $x=a$ ($a$ is constant).

Step 2: Simplification

$x=r\cos \theta ,y=r\sin \theta ,z=z$ are cylindrical coordinates.

Use $x=r\cos \theta$ in the given equation to convert equation to cylindrical coordinates.

$x=a$

$\Rightarrow r\cos \theta =a$

$r=\frac{a}{\cos \theta }$

$r=a\sec \theta$

Hence proved.