Problem 10 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r \cos \theta=-5\)

Step-by-Step Solution

Answer

The graph is a vertical line at \(x = -5\) in Cartesian coordinates.

Step 1 - Understand the given equation

The given equation is in polar coordinates: \(r \cos \theta = -5\). Here, \(r\) represents the radius, and \(\theta\) represents the angle in radians. In rectangular coordinates, this equation resembles the equation of a vertical line.

Step 2 - Convert Polar Equation to Rectangular Form

Using the conversion formula \(x = r \cos \theta\), we can rewrite the equation as \(x = -5\). This is now a familiar form of a vertical line in Cartesian coordinates.

Step 3 - Draw the Cartesian Equivalent

Sketch the vertical line \(x = -5\) on the rectangular coordinate system. This line passes through all points where the x-coordinate is -5.

Step 4 - Interpret in Polar Coordinates

In polar coordinates, this vertical line means the radius \(r\) can be any value, but the angle \(\theta\) must be such that \(r \cos \theta = -5\).

Step 5 - Draw the Graph in Polar Coordinates

In the polar coordinate system, this would look like a line that extends infinitely in both directions, passing through the point \((-5, \pi)\) on the negative x-axis.

Key Concepts

Polar to Rectangular Conversion

Polar coordinates use a different system to define the location of points. Instead of using horizontal and vertical distances from an origin (like in rectangular or Cartesian coordinates), polar coordinates use a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis).

A common task is converting between these two systems. The core formulas to convert from polar to rectangular coordinates are:

• x = r \, \cos \theta
• y = r \, \sin \theta

In our exercise, we were given the polar equation:

\(r \, \cos \theta = -5\).

By using the formula for x, we can rewrite this equation in rectangular form as:
\( x = -5 \).

This tells us that the x-coordinate of every point on this line is -5, simply a vertical line in Cartesian coordinates.

Vertical Line in Cartesian Coordinates

Understanding the geometry of a vertical line in Cartesian coordinates is crucial. In the rectangular coordinate system, a vertical line equation can be written as: \(x = c\), where \(c\) is a constant.

This means all points on this line have the same x-coordinate (\(c\)) but can have any y-coordinate value.

For the given problem, the polar equation: \(r \, \cos \theta = -5\) converts to the Cartesian equation: \(x = -5\). It tells us that no matter the value of \(y\) the x-coordinate is always -5. This line extends vertically through the entire y-axis at \cons{x = -5}.

Here's a quick rundown:

• The line crosses the x-axis at \( -5\).
• It goes vertically up and down, meaning it has no slope.

Graphing Equations

Graphing mathematical equations helps visualize mathematical concepts. Both polar and Cartesian systems have unique ways of representing points.

In Cartesian coordinates, graphing \( x = -5 \) is straightforward: draw a line through all points where the x-coordinate equals -5.

In polar coordinates, you interpret \( r \cos \theta = -5 \). The radius (\(r\)) can take any value, but the angle ( \( \theta \)) must satisfy the condition so that when multiplied by the cosine of \( \theta \), it equals -5.

Graphing this in polar coordinates means creating a line that matches this condition. In this case, the line extends infinitely in both directions through the point where \(r = -5\) and \(\theta = \pi\). Remember, in polar graphs, lines can be quite different from Cartesian lines, but fundamentally represent the same spatial relationships.

Summary:

• In Cartesian: a clear vertical line at \( x = -5\)
• In Polar: an infinite line meeting the condition \( r \cos \theta = -5 \)