Problem 23 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ 3 x-y+2=0 $$

Step-by-Step Solution

Answer

The polar form of the given rectangle equation is \(r = -2/(3\cos\theta - \sin\theta)\). The sketch involves a curve, with its maximum in the \(3\pi/4\) direction, extends indefinitely in the \(\pi/4\) and \(-3\pi/4\) directions. Where points are negative, reflect them across the origin.

Step 1: Convert Rectangular Equations into Polar form

In the given equation \(3x - y + 2 = 0\), replace \(x\) with \(r \cos \theta\) and \(y\) with \(r \sin \theta\). The equation becomes \(3r \cos \theta - r \sin \theta + 2 = 0\). To simplify further, factor out the \(r\) which yields \(r(3 \cos \theta - \sin \theta )+2=0\).

Step 2: Solve for r

For the graph, it is easier to have \(r\) on one side of the equation. Subtract 2 from both side to get \(r(3 \cos \theta - \sin\theta) = -2\). Then divide both sides by \((3 \cos \theta - \sin \theta)\) to isolate \(r\), producing \(r = -2/(3 \cos \theta - \sin \theta)\).

Step 3: Sketch the Graph

The graph sketching involves plotting the points of \(r\) and \(\theta\) on a polar grid. As \(\theta\) varies from \(0\) to \(360^{\circ}\) or \(0\) to \(2\pi\), mark the corresponding \(r\) values. Remember, \(r<0\) means the point is in the opposite direction. Properly interpret points where \(r\) is negative by reflecting over the origin and acquire smooth curve by joining these points.