Problem 58 - 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 \(\theta\). $$ 2 x^{2}+4 x y+2 y^{2}=9 $$

Step-by-Step Solution

Answer

The polar form is \(r(\theta) = \sqrt{\frac{9}{2}}\).

Step 1: Identify the general form of the conic

The given equation is \(2x^2 + 4xy + 2y^2 = 9\). It is in the form of a quadratic equation with an \(xy\) term, which suggests that the graph is rotated.

Step 2: Determine the rotation angle (\(\theta\))

For a conic equation \(Ax^2 + Bxy + Cy^2 = 0\), the rotation angle \(\theta\) can be calculated using \(\tan 2\theta = \frac{B}{A-C}\). Here, \(A = 2\), \(B = 4\), \(C = 2\). Since \(A = C\), \(\tan 2\theta = \frac{4}{0}\), which is undefined. This means \(\theta\) is \(45^\circ\) or \(\theta = \frac{\pi}{4}\) radians.

Step 3: Use the rotation formulas

To eliminate the \(xy\) term, set \(x = x'\cos\theta - y'\sin \theta\) and \(y = x'\sin\theta + y'\cos \theta\) where \(\theta = \frac{\pi}{4}\). Simplifying gives: \(x = \frac{1}{\sqrt{2}}(x' - y')\) and \(y = \frac{1}{\sqrt{2}}(x' + y')\). Substitute these into the original equation.

Step 4: Simplify to find the equation in rotated coordinates

Substitute and simplify \[(x')^2 + (y')^2 = \frac{9}{2}\]. This is the equation of a circle in the rotated coordinate system with radius \(\sqrt{\frac{9}{2}}\).

Step 5: Convert to polar coordinates

The polar form of a circle centered at the origin is \(r = \text{constant}\). Here, the radius \(r = \sqrt{\frac{9}{2}}\), hence in polar coordinates \(r(\theta) = \sqrt{\frac{9}{2}}\).

Key Concepts

Polar Coordinates

Polar coordinates provide a convenient way to describe the position of a point in a plane using a distance and an angle. Instead of using the Cartesian system with coordinates

• \(x\)
• \(y\)

polar coordinates use:

• \(r\) - the distance from the origin
• \(\theta\) - the angle measured from the positive x-axis

The relationships between Cartesian and polar coordinates are:\[x = r \cos \theta\]\[y = r \sin \theta\]These equations allow us to convert between these systems easily. When translating an equation like \(2x^2 + 4xy + 2y^2 = 9\) into polar coordinates, the main goal is to express \(r\) as a function of \(\theta\). This shift not only simplifies computations involving rotations but also provides an intuitive grasp on how geometric curves, like conics, behave in different orientations.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include familiar shapes like the

• circle,
• ellipse,
• parabola, and
• hyperbola.

These shapes occur naturally in many fields, such as astronomy and physics.

The given equation, \(2x^2 + 4xy + 2y^2 = 9\), is in the form of a conic with an \(xy\) term. This inclusion indicates that the graph is rotated. In this exercise, we specifically encountered a circle after transforming the coordinates. Identifying the different sections accurately allows us to better analyze and manipulate the equations according to our needs. By focusing on the nature of conic sections, we gain insights into paths of planetary motion and the optics of reflecting light, highlighting their practical importance in various domains.

Equation Transformation

Equation transformation involves changing the form of an equation to reveal or simplify its structure. In the given exercise:

• Start with transforming the conic equation \(2x^2 + 4xy + 2y^2 = 9\).
• Identify the rotation by calculating \(\theta\) using \(\tan 2\theta = \frac{B}{A-C}\).

This rotational transformation aligns the axes with the graph's orientation, eliminating the troublesome \(xy\) term. Here, since \(\tan 2\theta\) was undefined, the angle \(\theta\) resulted to be \(45^\circ\). Equation transformations, like rotation, are essential in simplifying and solving complex mathematical problems, especially when dealing with graphs that do not align neatly with the standard axes.

Trigonometric Conversion

Trigonometric conversion is a toolkit that helps in modifying equations by using trigonometric identities. When performing a rotation:

• We use substitution formulas, such as \(x = x'\cos\theta - y'\sin\theta\) and \(y = x'\sin\theta + y'\cos\theta\),
• where \(\theta\) is the angle of rotation, here \(\theta = \frac{\pi}{4}\) or 45 degrees.

This conversion removed the \(xy\) term, leading to the simplified equation \((x')^2 + (y')^2 = \frac{9}{2}\). By understanding and using these trigonometric relationships, one can maneuver through complex mathematical landscapes with ease. Trigonometric conversion not only assists in rewriting equations but also deepens our understanding of how angles and distances interact in geometry and physics.