Problem 75 - Algebra and Trigonometry

Question

Show that each statement is true by converting the given polar equation to a rectangular equation. Show that the graph of \(r=a \sec \theta\) is a vertical line \(a\) units to the right of the \(y\) -axis if \(a>0\) and \(|a|\) units to the left of the \(y\) -axis if \(a<0\)

Step-by-Step Solution

Answer

After converting the polar equation \(r = a \sec \theta\) into the rectangular form, we find that \(x = a\), which is a vertical line at a distance 'a' from the y-axis. Therefore, the exercise statement about the vertical line is valid.

Step 1: Understand Polar Coordinates

Polar coordinates (r, θ) can be converted into rectangular coordinates (x, y) by using the following relationship: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Convert Polar to Rectangular Coordiantes

Replace the polar equation \(r=a \sec \theta\) with rectangular equations, i.e., replace r with \(x = a\cos \theta\), gives \(x = a^2 \cos \theta\).

Step 3: Simplify the equation

Simplify the equation \(x=a^2 \cos \theta\) to \(x= a\), which is the equation of a vertical line.

Step 4: Verify statement

For \(a > 0\), the line \(x = a\) is to the right of the y-axis and for \(a < 0\), \(x = |a|\) is to the left of the y-axis.