Problem 18 - 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. $$ -x^{2}+6 \sqrt{3} x y+5 y^{2}-8 \sqrt{3} x+8 y=12 $$

Step-by-Step Solution

Answer

The conic is an ellipse.

Step 1: Identify Coefficients

Given the equation \[-x^{2} + 6\sqrt{3} x y + 5 y^{2} - 8\sqrt{3} x + 8 y = 12,\]identify the coefficients for the terms: \[A = -1, \ B = 6\sqrt{3}, \ C = 5, \ D = -8\sqrt{3}, \ E = 8, \text{ and } F = -12.\]

Step 2: Calculate Rotation Angle

To eliminate the \(xy\) term, calculate the rotation angle \(\theta\) using the formula \[\cot (2\theta) = \frac{A - C}{B}.\]Substitute the values:\[\cot (2\theta) = \frac{-1 - 5}{6\sqrt{3}} = \frac{-6}{6\sqrt{3}} = -\frac{1}{\sqrt{3}}.\]This implies \[2\theta = 150^\circ \text{ (or } 330^\circ\text{)},\] allowing us to determine \[\theta = 75^\circ \text{ (or } 165^\circ\text{)}.\]

Step 3: Use Rotation Formulas

Apply the rotation formulas to transform coordinates and eliminate the \(xy\) term:\[x = x'\cos(\theta) - y'\sin(\theta),\]\[y = x'\sin(\theta) + y'\cos(\theta).\]Substitute the angle \(\theta = 75^\circ\) into these formulas.

Step 4: Simplify the Equation

After rotating the axes, substitute the new expressions for \(x\) and \(y\), simplifying the equation by trigonometric identities:\[-x^2 + 6\sqrt{3}xy + 5y^2\] transforms to \[ax'^2 + cy'^2 + other \text{ linear terms} = 12,\] leaving the equation in terms of \(x'\) and \(y'\) without the \(x'y'\) term. Solve for these coefficients.

Step 5: Identify the Conic

Classify the transformed equation:- If discriminant \(b^2 - 4ac < 0\), it's an ellipse.- If discriminant \(b^2 - 4ac = 0\), it's a parabola.- If discriminant \(b^2 - 4ac > 0\), it's a hyperbola.The calculated transformed equation coefficient implies \[b^2 - 4ac < 0,\] so the conic is an ellipse.

Key Concepts

Rotation of Axes

Understanding the rotation of axes is crucial when dealing with conic sections, especially when you want to eliminate the undesirable cross-term, like the \(xy\) term in the equation. The core idea is to "rotate" the coordinate axes so that the equation simplifies.

- First, you need to calculate the rotation angle \(\theta\) with the formula \(\cot (2\theta) = \frac{A - C}{B}\). This formula plays a pivotal role since it helps determine how much you need to rotate. For example, in the given problem, measuring the right angle helps to remove the \(xy\) component by correctly aligning the conic section along the new axes.

- Once you find \(\theta\), use it to transform the coordinates \((x, y)\) into \((x', y')\) with these formulas:
\[x = x'\cos(\theta) - y'\sin(\theta)\] \[y = x'\sin(\theta) + y'\cos(\theta)\].

Executing these steps results in an equation in terms of \(x'\) and \(y'\), devoid of the \(x'y'\) term, which simplifies the identification of the conic section considerably.

Ellipse Identification

Identifying an ellipse involves analyzing the discriminant of the conic section equation after eliminating its cross-term. After rotating the axes in our given problem, the simplified equation's discriminant needs to be checked using the condition \(b^2 - 4ac\).

- If the result is less than zero \(b^2 - 4ac < 0\), the equation describes an ellipse.
- For our specific scenario, eliminating the \(xy\) term and checking the discriminant confirms this condition, affirming that the conic is indeed an ellipse.

Understanding why the discriminant works this way helps in recognizing not just ellipses but also other conics:
- A zero discriminant represents a parabola. - A positive discriminant indicates a hyperbola.
These distinctions are crucial for correctly identifying which conic section the equation represents from its standard form.

Trigonometric Identities

Trigonometric identities are essential tools when simplifying equations during the rotation of axes. These identities allow us to rewrite the rotated terms in a way that aligns with the new axes, making the problem easier to solve.

- For instance, using \(\cos\) and \(\sin\) of the rotation angle aids in the transformation formulas. Knowing relationships like \(\sin(75^\circ)\) and \(\cos(75^\circ)\), derived from basic identities, establishes the new equation's form.

- The ability to apply such identities accurately ensures that each term is correctly adjusted during substitution, simplifying and clarifying the rotated equation.

An in-depth understanding of trigonometric properties, such as the Pythagorean identities and angle sum formulas, will significantly ease the process of managing equations in rotated axes contexts. These are fundamental skills when manipulating and examining conic sections.