Problem 59 - 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 \(\theta\). $$ 16 x^{2}+24 x y+9 y^{2}=4 $$

Step-by-Step Solution

Answer

The polar form is \(r = \frac{2}{4 \cos \theta + 3 \sin \theta}\).

Step 1: Identify the equation form

Given equation is in the standard form: \(16x^{2}+24xy+9y^{2}=4\). This is a quadratic equation in two variables \(x\) and \(y\).

Step 2: Apply polar coordinate transformation

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Substitute these into the equation to get: \[16(r \cos \theta)^2 + 24(r \cos \theta)(r \sin \theta) + 9(r \sin \theta)^2 = 4\].

Step 3: Simplify the equation

Expand each term: \((r \cos \theta)^2 = r^2 \cos^2 \theta\), \((r \sin \theta)^2 = r^2 \sin^2 \theta\), and combine terms: \(16r^2 \cos^2 \theta + 24r^2 \cos \theta \sin \theta + 9r^2 \sin^2 \theta = 4\). Factor out \(r^2\): \[r^2 (16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta) = 4\].

Step 4: Solve for \(r^2\)

The expression in parentheses simplifies to: \((16 \cos^2 \theta + 24 \cos \theta \sin \theta + 9 \sin^2 \theta)\) matches a perfect square: \((4 \cos \theta + 3 \sin \theta)^2\). Therefore, \[r^2 (4 \cos \theta + 3 \sin \theta)^2 = 4\].

Step 5: Express \(r\) as a function of \(\theta\)

Divide both sides by \((4 \cos \theta + 3 \sin \theta)^2\) to isolate \(r^2\): \[r^2 = \frac{4}{(4 \cos \theta + 3 \sin \theta)^2}\]. Take the square root to find \(r\): \[r = \frac{2}{4 \cos \theta + 3 \sin \theta}\].

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. The primary shapes formed are circles, ellipses, parabolas, and hyperbolas. Each shape has distinct features based on the plane's angle and position relative to the cone. This unique property allows conic sections to be described by quadratic equations. These quadratic forms can represent certain conics; for instance, a circle has an equation without an "xy" term, while other conics like the hyperbola and the ellipse might have this term. In polar coordinates, these equations can exhibit different properties that facilitate the study of their geometry and transformation behaviors.

Rotation of Axes

In algebra, the rotation of axes is a technique used to simplify equations with terms like "xy." When facing such equations, by rotating the coordinate system, the "xy" term can often be eliminated. This makes it easier to recognize the conic section represented by the equation.

• This technique is particularly valuable for equations like the given one, \(16x^{2}+24xy+9y^{2}=4\), where the "xy" term signifies that the graph is tilted.
• By rotating the axes, you align the new axes with the principal axes of the conic, simplifying the equation into a standard form easier to analyze.

Understanding rotation assists in revealing the natural orientation and properties of the conic without interference from the original coordinate orientation.

Quadratic Equations

Quadratic equations in two variables produce conic sections. The general form is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). The presence of the "xy" term typically makes the conic tilted in the plane. Quadratics help describe the size, position, and shape of conics:

• The signs and magnitude of coefficients \(A, B,\) and \(C\) influence the conic's type.
• In our specific exercise, the equation \(16x^2 + 24xy + 9y^2 = 4\) is manipulated using polar coordinates, which aids in unearthing its underlying conical shape by looking at it through a different lens, or coordinate system.

These equations are foundational in understanding more complex functions and transformations in mathematics.

Transformation

Transformation is a method used to change the view or form of an equation or geometric shape. In the context of conic sections and polar coordinates, transformation facilitates better understanding and interpretation of equations:

• Polar coordinate transformation transitions between rectangular \((x, y)\) and polar \((r, \theta)\) forms.
• This transformation unveils different perspectives of the same shape, as was done in the original problem by substituting \(x = r \cos \theta\) and \(y = r \sin \theta\).
• It often simplifies the mathematics involved and allows for clearer visualization, making the properties of the conic more accessible.

By transitioning to polar coordinates and addressing the equation \(16x^2 + 24xy + 9y^2 = 4\), the complex relationship becomes manageable as we isolate \(r\), expressing it solely in terms of \(\theta\). This is powerful, as it redefines geometry in terms of circular and radial components.