Problem 28 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=-2 $$

Step-by-Step Solution

Answer

The rectangular form of the given equation \(-2\), is \(x^2 + y^2 = 4\), which represents a circle with a radius of 2 centered at the origin. The graph is a circle reflected across the origin.

Step 1: Convert Polar Equation to a Rectangular Form

Recall that the polar coordinates are related to the rectangular coordinates through the formulas: \(x=r \cos(\theta)\) and \(y=r \sin(\theta)\). The given polar equation is \(r = -2\). Insert this into the equations for the conversion. Since \( r = -2 \), we can replace \( r \) in the conversion formulas directly to get \(x = -2 \cos(\theta)\) and \(y = -2 \sin(\theta)\).

Step 2: Solving for the Rectangular Coordinate

Although we have equations for x and y, a single equation is needed that relates x and y to draw a graph in rectangular form. Since we know that \(x^2 + y^2 = r^2\) in polar coordinates, substitute \( r = -2 \) into this equation to get \(x^2 + y^2 = (-2)^2 = 4\). This is the equation of a circle centered at the origin with a radius of 2 in rectangular coordinates.

Step 3: Sketch the Graph

Draw a circle centered at the origin with a radius of 2. Since the initial r-value given was negative, indicating a direction opposite from the usual direction, the actual graph would be the reflection of the circle across the origin.