Problem 25 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ y^{2}=9 x $$

Step-by-Step Solution

Answer

The polar form of the equation is \(r\sin^2(\theta) = 9\cos(\theta)\) and the graph is undefined at \(\theta = 0\) with parabolic symmetry.

Step 1: Conversion of Variables

In polar coordinates, the x and y variables of the rectangular equation can be replaced with \(r\cos(\theta)\) and \(r\sin(\theta)\) respectively. Therefore, the given rectangular equation: \(y^2 = 9x\) can be re-written as \((r\sin(\theta))^2 = 9(r\cos(\theta))\) which simplifies to \(r^2\sin^2(\theta) = 9r\cos(\theta)\).

Step 2: Simplify the Polar Equation

Rewrite the equation from step 1 by canceling out the common parameter r from both sides giving \(r\sin^2(\theta) = 9\cos(\theta)\). We can only cancel the common r if \(r \neq 0\). This is an essential step to ensure there are no mathematical fallacies in the solution.

Step 3: Graphing the Polar Function

To graph this function, observe that the function will have values when the right side is equal to the left side. Hence, the equation represents a graph where for every theta, r equals to \(9\cos(\theta)\) divided by \(\sin^2(\theta)\) provided \(\theta \neq 0\). This graph is undefined at \(\theta = 0\), displays symmetry, and will have a parabolic shape.