Problem 57 - College Algebra

Question

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ x^{2}+x y+y^{2}=4 $$

Step-by-Step Solution

Answer

\(r = \sqrt{\frac{4}{1 + \cos\theta \sin\theta}}\)

Step 1: Identify the Conic Equation

The given equation is \(x^2 + xy + y^2 = 4\). This is a rotated conic since it includes an \(xy\) term.

Step 2: Understand Rotation Terms

A conic with an \(xy\) term indicates rotation. We need to eliminate this term by rotating the axes or directly convert to polar coordinates.

Step 3: Convert to Polar Coordinates

Use the polar coordinates transformations: \(x = r \cos\theta\) and \(y = r \sin\theta\). Substitute these into the equation.

Step 4: Substitute in the Equation

Replace \(x\) and \(y\) in the equation: \((r \cos\theta)^2 + (r \cos\theta)(r \sin\theta) + (r \sin\theta)^2 = 4\).

Step 5: Simplify the Expression

Expand the expression: \(r^2 \cos^2\theta + r^2 \cos\theta \sin\theta + r^2 \sin^2\theta = 4\). Factor out \(r^2\).

Step 6: Factor Out \(r^2\)

Recognize that \(r^2(\cos^2\theta + \sin^2\theta + \cos\theta \sin\theta) = 4\). Use the identity \(\cos^2\theta + \sin^2\theta = 1\).

Step 7: Further Simplify the Equation

The equation becomes \(r^2(1 + \cos\theta \sin\theta) = 4\).

Step 8: Express \(r\) as a Function of \(\theta\)

Solve for \(r^2\): \(r^2 = \frac{4}{1 + \cos\theta \sin\theta}\). Hence, \(r = \sqrt{\frac{4}{1 + \cos\theta \sin\theta}}\).

Key Concepts

Rotated Conics

Rotated conics are fascinating because they include terms that involve both the variables together, like the xy-term in equations. This unique term means that the conic is not aligned with the standard x or y axes. It suggests that to fully understand the shape, one must consider an axis rotation.
By rotating the axes, the conic can be aligned with these new axes, often simplifying the expression by eliminating the xy-term. This transformation reveals the true nature of the conic. In its standard position, without the xy-term, it is easier to classify and study the conic effectively.
The given original equation, with an xy-term, is a prime example of such a rotated conic. Using mathematical transformations, like converting to polar coordinates, helps to explore these shapes in a different and often simpler context.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. The primary types include:

• Circles: Formed when the intersecting plane is perpendicular to the cone's axis.
• Ellipses: Occur when the plane cuts through the cone at an angle but does not run parallel to the sides.
• Parabolas: Created when the plane is parallel to the length of the cone.
• Hyperbolas: Formed when the plane intersects both halves of the double cone.

Conic sections are vital in various fields including astronomy, physics, and engineering because they describe the paths of celestial bodies, design of reflective surfaces, and more. Recognizing a conic section in an equation involves examining the coefficients and terms present, especially noting any rotations indicated by the presence of xy terms, as seen in the original exercise conic equation. Each type of conic has distinct properties and mathematical characteristics that define their equations and graphs.

Coordinate Transformation

Coordinate transformation is a method of redefining the coordinate system to simplify equations or to better understand geometrical shapes. One of the common transformations involves switching from Cartesian coordinates \(x, y\) to polar coordinates \(r, \theta\).
Polar coordinates describe a point's position based on its distance from the origin \(r\), and the angle \(\theta\) from the positive x-axis. To convert from Cartesian to polar, use transformations: \(x = r \cos\theta\) and \(y = r \sin\theta\).
This is particularly useful in rotated conics where using polar coordinates can simplify the analysis by reducing complex equations. In our example, by switching to polar coordinates, the equation simplifies, making it easier to solve for \(r\) as a function of \(\theta\). This technique is essential in fields such as physics and engineering, where dealing with circular or rotational symmetries is common.