Problem 59 - College Algebra with Corequisite Support

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 \(r = \sqrt{\frac{4}{16\cos^2\theta + 24\cos\theta \sin\theta + 9\sin^2\theta}}.\)

Step 1: Identify the general conic equation

The given equation is in the form of a general conic section which can include a rotation due to the presence of the \(xy\) term: \(16x^2 + 24xy + 9y^2 = 4\). Our task is to express this in polar coordinates.

Step 2: Express in terms of polar coordinates

We know the relationships \(x = r\cos\theta\) and \(y = r\sin\theta\). Substitute these into the equation: \[16(r\cos\theta)^2 + 24(r\cos\theta)(r\sin\theta) + 9(r\sin\theta)^2 = 4.\]

Step 3: Simplify the equation

Expand the equation: \[16r^2 \cos^2\theta + 24r^2 \cos\theta \sin\theta + 9r^2 \sin^2\theta = 4.\] Factor out \(r^2\), resulting in: \[r^2(16\cos^2\theta + 24\cos\theta \sin\theta + 9\sin^2\theta) = 4.\]

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

Divide every term by 4 to separate \(r^2\): \[r^2 = \frac{4}{16\cos^2\theta + 24\cos\theta \sin\theta + 9\sin^2\theta}.\]

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

Take the square root of both sides to solve for \(r\): \[r = \sqrt{\frac{4}{16\cos^2\theta + 24\cos\theta \sin\theta + 9\sin^2\theta}}.\] This represents the polar form of the equation.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double napped cone. The common conic sections include circles, ellipses, parabolas, and hyperbolas. They have unique algebraic equations that describe their geometric shapes. These equations generally take the form:

• Circle: \(x^2 + y^2 = r^2\)
• Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)
• Parabola: \(y = ax^2 + bx + c\)
• Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

In some cases, as seen in the equation \(16x^2 + 24xy + 9y^2 = 4\), conic sections can have an \(xy\) term, indicating the graph is rotated. This can imply a transformation in its usual coordinate axes.Recognizing whether an \(xy\) term exists is crucial to identify conic sections with rotated axes, as it influences the orientation of the conic in a fixed coordinate plane. Once identified, these equations often require conversion into other coordinate systems, such as polar coordinates, for further analysis.

Rotation of Axes

The rotation of axes is a mathematical technique used to simplify equations, especially those that involve an \(xy\) term. The presence of this term suggests that the graph of the equation is rotated relative to the standard coordinate axes.For such equations, it is often useful to apply a change of variables that involves trigonometric identities:

• \(x = X\cos\theta - Y\sin\theta\)
• \(y = X\sin\theta + Y\cos\theta\)

This transformation removes the \(xy\) term by correctly choosing \(\theta\), the angle of rotation, thus aligning the graph into a standard position either vertically or horizontally. This process helps to make the equation simpler and easier to handle.In our problem, we've skipped this classical rotation step to directly explore conversion to polar form, which inherently simplifies the coordinate system by using radius \(r\) and angle \(\theta\) instead of \(x\) and \(y\) coordinates, thus dealing with the inherent rotation implicitly.

Converting to Polar Form

Converting a Cartesian equation to its polar form involves using the relationships between Cartesian coordinates \(x, y\) and polar coordinates \(r, \theta\). In polar coordinates:

• Encrypt \(x = r\cos\theta\)
• Encrypt \(y = r\sin\theta\)

Substitution into the original equation allows us to replace \(x\) and \(y\) with expressions involving \(r\) and \(\theta\). For example, substituting into the equation \(16x^2 + 24xy + 9y^2 = 4\), we applied these relationships and simplified step-by-step to find \(r\) as a function of \(\theta\).This process helps in analyzing the behavior of the conic sections in systems where the angle from a reference direction is more informative or natural, such as in circular or rotational systems.The final equation in polar form, \(r = \sqrt{\frac{4}{16\cos^2\theta + 24\cos\theta \sin\theta + 9\sin^2\theta}}\), shows how the radius changes as the angle \(\theta\) changes, providing a complete angular perspective of the conic section's graph. This form is especially useful for calculus problems involving derivatives and integrals in polar coordinates.