Problem 21 - Precalculus with Calculus Previews

Question

In Problems \(17-20,\) use rotation of axes to eliminate the \(x y\) -term in the given equation. Identify the conic. Given \(3 x^{2}+2 \sqrt{3} x y+y^{2}+2 x-2 \sqrt{3} y=0\) (a) By rotation of axes show that the graph of the equation is a parabola. (b) Find the \(x^{\prime} y^{\prime}\) -coordinates of the focus. Use this information to find the \(x y\) -coordinates of the focus. (c) Find an equation of the directrix in terms of the \(x^{\prime} y^{\prime}\) -coordinates. Use this information to find an equation of the directrix in terms of the \(x y\) -coordinates.

Step-by-Step Solution

Answer

The graph is a parabola; focus and directrix are translated via transformation.

Step 1: Identify the Rotation Angle

The angle \( \theta \) for rotation of axes to eliminate the \( xy \)-term can be found using \( \tan(2\theta) = \frac{B}{A-C} \), where \( A = 3 \), \( B = 2 \sqrt{3} \), and \( C = 1 \). So, \( \tan (2\theta) = \frac{2\sqrt{3}}{2} = \sqrt{3} \). Therefore, \( \theta = 30^\circ \).

Step 2: Form the New Variables Using Rotation

The new variables \( x' \) and \( y' \) are found using: \[ x = x' \cos \theta - y' \sin \theta \quad \text{and} \quad y = x' \sin \theta + y' \cos \theta \]Substitute \( \theta = 30^\circ \) to get: \[ x = \frac{\sqrt{3}}{2} x' - \frac{1}{2} y' \quad \text{and} \quad y = \frac{1}{2} x' + \frac{\sqrt{3}}{2} y' \]

Step 3: Substitute New Variables into the Original Equation

Plug the expressions for \( x \) and \( y \) in terms of \( x' \) and \( y' \) into the original equation and simplify the terms. This will eliminate the \( x'y' \)-term, yielding:\[ 4x'^2 + (x' - \sqrt{3}y') + y' = 0 \]

Step 4: Identify the Conic Section

After removing the \( x'y' \)-term, the resulting equation \[ 4x'^2 + (x' - \sqrt{3}y') + y' = 0 \] is in the form of a parabola, indicating that the original equation is a parabola.

Step 5: Find x'y'-coordinates of the Focus

The equation can be rewritten in the standard parabola form \( y'^2 = 4px' \) to evaluate \( p \). Rewriting gives \( y' = -(4x'^2 + (x' - \sqrt{3}y')) \), identifying focus at \((p,0)\), calculate \(p\) using the coefficient.

Step 6: Convert Focus Coordinates to Original x,y

Using the inverse of the rotation transformation, \( x = x' \cos \theta + y' \sin \theta \) and \( y = -x' \sin \theta + y' \cos \theta \), convert the \(x'y'\)-coordinates of the focus back to \(x,y\)-coordinates.

Step 7: Find the Equation of the Directrix

In \( x'\)-\( y'\) coordinates, the directrix is at \( x' = -p \). Convert this back to \( x, y\) coordinates using inverse rotation equations to get the directrix equation in terms of \( x \) and \( y \).

Key Concepts

Rotation of Axes

The rotation of axes is a powerful technique used in geometry and algebra to simplify the analysis of conic sections by eliminating the cross-term, often represented as the \(xy\)-term, in a quadratic equation.
To achieve this, we rotate the coordinate system by an angle \(\theta\). This rotation makes it easier to identify and study individual conic sections like ellipses, hyperbolas, and parabolas.

• Calculate the angle \(\theta\) to eliminate the \(xy\)-term using the equation \(\tan(2\theta) = \frac{B}{A-C}\), where \(A\), \(B\), and \(C\) are coefficients from the quadratic equation.
• The formulas for the new rotated coordinates \(x'\) and \(y'\) are given by:
  \( x = x' \cos \theta - y' \sin \theta \)
• \( y = x' \sin \theta + y' \cos \theta \)

By substituting these into the original equation, the \(xy\)-term is eliminated, making the equation easier to analyze.

Parabola

A parabola is one of the most common conic sections and has a distinct U-shaped curve. It is defined as the set of all points equidistant from a fixed point known as the "focus" and a line known as the "directrix."

A standard form equation of a parabola is either \(y^2 = 4px\), \(x^2 = 4py\) or variations where \(p\) is the distance from the vertex to the focus or directrix. Here are some features of a parabola:

• The vertex is the turning point of the parabola.
• The axis of symmetry is a line that passes through the vertex and focus, dividing the parabola into two mirror-image halves.
• The focus is a point inside the parabola where all points reflect off the parabola and converge.

Identifying a parabola in a conic equation often involves recognizing its standard form after removing any crossed variable terms.

Focus and Directrix

The focus and directrix are fundamental components of a parabola.

• The focus of a parabola is a point from which the distance to any point on the parabola is equal to the perpendicular distance from that point to the directrix.
• The directrix is a line parallel to the parabola's axis of symmetry but external to it, contributing to the parabola's defined shape.
• These components are used to derive the parabolic path, ensuring all points on the parabola maintain the true parabolic condition.

To find these components:
1. Identify the standard form of the parabolic equation.
2. Rearrange the equation where possible to recognize the focus and directrix clearly.
This will help in understanding the parabola's geometry in both new \((x',y')\) and original \((x,y)\) coordinates.

Coordinate Transformation

Coordinate transformation is key when altering the position or orientation of a conic section in the Cartesian plane.

• This is done by changing the original \((x, y)\) coordinates to a new set \((x', y')\) that makes the analysis of the conic section more straightforward.
• With transformation formulas like \(x = x' \cos \theta - y' \sin \theta\) and \(y = x' \sin \theta + y' \cos \theta\), we can transfer this understanding by rotating the axes by angle \(\theta\).

Post-transformation, the conic section's equation is simplified, making it easier to identify important elements such as the type of conic, its center, vertices, focus, and directrix.
Once the analysis is done, it's crucial to transform back, ensuring the correct interpretation in the original coordinate system. This back transformation is done using inverse formulas, keeping the study both comprehensive and practical.