Problem 47 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r^{2}\left(4 \sin ^{2} \theta-9 \cos ^{2} \theta\right)=36$$

Step-by-Step Solution

Answer

The Cartesian equation is \(4y^2 - 9x^2 = 36\), a hyperbola.

Step 1: Rewrite the Polar Equation

The given polar equation is \(r^{2}(4\sin^{2}\theta - 9\cos^{2}\theta) = 36\). Our aim is to convert this into a Cartesian equation in terms of \(x\) and \(y\). In polar coordinates, \(r^{2} = x^{2} + y^{2}\), \(r\sin\theta = y\), and \(r\cos\theta = x\). Let's express \(r^2\sin^2\theta\) and \(r^2\cos^2\theta\) using these relationships.

Step 2: Express Components in Cartesian Variables

\(r^{2}\sin^{2}\theta = y^{2}\) and \(r^{2}\cos^{2}\theta = x^{2}\). Therefore, plugging into the equation we have: \((4y^{2} - 9x^{2}) = 36\).

Step 3: Simplify the Cartesian Equation

Now, solve for the equation in a more recognizable form: \[4y^{2} - 9x^{2} = 36.\]This equation is of the form \(Ax^2 + By^2 = C\), which suggests it is a conic section, specifically a hyperbola.

Step 4: Identify and Sketch the Conic

We recognize \(4y^2 - 9x^2 = 36\) as a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Rewriting it as \( \frac{9x^2}{36} - \frac{4y^2}{36} = -1\), gives us this form. Thus, it represents a hyperbola with transverse axis along the \(y\)-axis. Sketch this hyperbola in the \(xy\)-plane then translate this understanding to the \(r\theta\)-plane.

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing a point in a plane using a radius and an angle. Unlike the more common Cartesian coordinates, which use an x and y value, polar coordinates use a distance from a particular point (usually the origin, denoted as 'r') and an angle (usually measured from the positive x-axis, denoted as '\( \theta \)').
This system makes it easier to handle problems with circular or rotational symmetry.Some core aspects include:

• The radius '\( r \)' is the distance from the origin to the point.
• The angle '\( \theta \)' is the angle between the positive x-axis and the line connecting the origin to the point.

These two values give the exact location of a point. This system is particularly useful in navigation and physics where rotational motion often occurs.

Cartesian Coordinates

Cartesian coordinates refer to the system where each point in a plane is identified by a pair of numbers (x, y). Each number represents a particular distance from two fixed perpendicular axes. This system is named after the French mathematician René Descartes.A few essential points about Cartesian coordinates are:

• '\( x \)' represents the horizontal distance from the y-axis.
• '\( y \)' measures the vertical distance from the x-axis.

This system is straightforward and widely used in tasks ranging from simple graph plotting to complex engineering calculations. When converting from polar to Cartesian coordinates, we use the formulas \( x = r\cos\theta \) and \( y = r\sin\theta \), making it easier to transition between different systems and solve problems accordingly.

Hyperbola

A hyperbola is a type of conic section formed by intersecting a double cone with a plane in a way that produces an open curve. Unlike a circle or ellipse, a hyperbola consists of two disconnected curves called branches.Here are a few features of hyperbolas:

• Defined by the difference of distances to two fixed points (foci) being constant for any point on the curve.
• Equations of hyperbolas often appear as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

In the provided step-by-step solution, we encountered a transformed hyperbola equation, \( 4y^2 - 9x^2 = 36 \), which maps out a hyperbola in the Cartesian plane. Recognizing and converting these equations from polar coordinates can help in interpreting and sketching curves in both mathematical and practical applications.