Problem 30 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=5 \cos \theta $$

Step-by-Step Solution

Answer

The polar equation \( r = 5\cos\theta \) converts to the rectangular form \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \), which represents a circle with radius 2.5 centered at \( ({5 \over 2}, 0) \).

Step 1: Converting the polar equation to rectangular form

We start with the polar equation \( r = 5\cos\theta \). Multiplying both sides of the equation by r yields \( r^2 = 5r\cos\theta \). We then substitute \( r = \sqrt{x^2 + y^2} \) and \( \cos\theta = {x \over r} \) into the equation to get \( x^2 + y^2 = 5x \). Rearranging terms, we have \( x^2 - 5x + y^2 = 0 \). Completing the square on x results in the rectangular equation \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \).

Step 2: Sketching the graph

The rectangular equation \( (x - {5 \over 2})^2 + y^2 = ({5 \over 2})^2 \) describes a circle with radius \( {5 \over 2} \) and center at \( ({5 \over 2}, 0) \). So, the sketch should look like a circle at point 2.5 on the x-axis and with radius 2.5