Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: 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 equivalent Cartesian equation is \( 9x^2 - 4y^2 = -36 \), representing a hyperbola.

Step 1: Identify known polar to Cartesian conversions

In polar coordinates, we know that \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r^2 = x^2 + y^2 \). These identities will help transform the given equation from polar to Cartesian coordinates.

Step 2: Simplify the given polar equation

The given polar equation is \( r^{2}(4 \sin^{2} \theta - 9 \cos^{2} \theta) = 36 \). This can be rewritten as \( r^2 (4 - 4\cos^2\theta - 9\cos^2\theta) = 36 \), which simplifies to \( r^2 (4 - 13\cos^2\theta) = 36 \).

Step 3: Express \( \sin^2 \theta \) and \( \cos^2 \theta \) in terms of \( x \) and \( y \)

Using the identities from Step 1: \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \), so \( \sin^2 \theta = \frac{y^2}{r^2} \) and \( \cos^2 \theta = \frac{x^2}{r^2} \). Let's substitute \( \cos^2 \theta \) in Step 2.

Step 4: Substitute and rearrange the equation

Now, replace \( \cos^2 \theta \) with \( \frac{x^2}{r^2} \) in \( r^2(4 - 13\cos^2\theta) = 36 \). This becomes \( r^2(4 - 13\frac{x^2}{r^2}) = 36 \). Simplifying gives \( 4r^2 - 13x^2 = 36 \) which is rearranged to \( 4(x^2 + y^2) - 13x^2 = 36 \).

Step 5: Complete the transformation to Cartesian coordinates

Continuing from Step 4, simplify further: \( 4x^2 + 4y^2 - 13x^2 = 36 \), resulting in \( -9x^2 + 4y^2 = 36 \). Rearranging, we get \( 9x^2 - 4y^2 = -36 \). This equation represents a hyperbola in Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using two values: the radius and the angle. These coordinates are denoted as \(r, \theta\). The radius \(r\) represents the distance from the origin (center point), while the angle \(\theta\) is measured from the positive x-axis.
This method is often used in scenarios where the relationship between the points is circular or spins around a central point.

• The origin is sometimes referred to as the pole in polar coordinates.
• The angle is typically measured in radians or degrees.

Understanding polar coordinates can make solving complex circular problems easier and provides an alternative to traditional Cartesian coordinates.

Cartesian Coordinates

Cartesian coordinates are the most common coordinate system used in mathematics and engineering. They specify each point uniquely in a plane by a pair of numerical coordinates: \(x\) and \(y\).
These coordinates denote positions in a grid-like layout using perpendicular axes.

• The x-coordinate represents horizontal positioning along the x-axis.
• The y-coordinate represents vertical positioning along the y-axis.
• They are very effective for most algebraic operations and geometrical plotting.

Cartesian coordinates are straightforward and ideal for linear equations and rectangles. However, they might not be as efficient as polar coordinates in dealing with circular trends or rotations.

Equation Transformation

Converting between different coordinate systems, like polar and Cartesian, is known as equation transformation. Polar to Cartesian transformations are essential for simplifying equations and for visualizing graphs in the xy-plane.
To convert polar coordinates to Cartesian:

• Use \(x = r \cos \theta\)
• Use \(y = r \sin \theta\)
• Understand that \(r^2 = x^2 + y^2\)

This set of conversions allows representation of the same equation in a different coordinate system. Transforming equations helps visualize relationships differently, depending on the problem or graph requirements.

Hyperbola

A hyperbola is a type of conic section that appears when a plane cuts through both halves of a double cone. It is a set of all points where the difference of the distances to two fixed points (foci) is a constant.
In Cartesian coordinates, a hyperbola typically exhibits an equation of the form:

• Standard: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Or its variations based on orientation and centered arrangement.

The equation derived from the given problem is \(9x^2 - 4y^2 = -36\), a hyperbola represented in Cartesian coordinates.
This mathematical function is crucial in fields like physics and engineering to describe systems with two routes or asymptotic behavior, often seeing applications like in optics and orbital mechanics.