Problem 94 - Algebra and Trigonometry

Question

In converting \(r=5\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

Step-by-Step Solution

Answer

The polar equation \(r=5\) is converted to the rectangular equation \(x^2 + y^2 = 25\). This is done by squaring both sides of the given equation and replacing \(r^2\) with \(x^2 + y^2\). This converts the equation to rectangular form and the result represents a circle of radius 5, centered at the origin.

Step 1: Understand the Conversion Formula

In order to convert polar coordinates to rectangular coordinates, it's important to understand the relationship between them. If we consider a point with polar coordinates (r, θ), where r is the distance from the origin and θ is the angle of the point from the positive x-axis, in a Cartesian system the coordinates of this point would be \((x, y)\), and the relationship between \((r, θ)\) and \((x, y)\) can be represented by \(x^2 + y^2 = r^2\).

Step 2: Apply the Conversion Formula

To start the conversion process, the given equation \(r = 5\) should be squared on both sides. This gives \(r^2 = 25\). Now, using the conversion formula previously mentioned, replace \(r^2\) with \(x^2 + y^2\). This gives \(x^2 + y^2 = 25\).

Step 3: Interpret the Result

The result, \(x^2 + y^2 = 25\), represents a circle in rectangular coordinates with a radius of 5 (since the radius r is the square root of 25) centered at the origin (0,0). This is the rectangular form of the polar equation \(r=5\).