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Chapter 10: Problem 5

Rotate the coordinate axes to remove the \(x y\) -term. Then identify the type of conic and sketch its graph. $$ x^{2}+4 x y-2 y^{2}-6=0 $$

Short Answer

Expert verified
After rotation, the conic is a hyperbola.

Step by step solution

01

Identify coefficients for rotation

The given equation is \( x^2 + 4xy - 2y^2 - 6 = 0 \). Coefficients for rotation are \( A = 1 \), \( B = 4 \), and \( C = -2 \). Since \( B eq 0 \), rotation is needed to eliminate the \( xy \)-term.
02

Use rotation formulas

Find the angle \( \theta \) for rotation using \( \tan(2\theta) = \frac{B}{A - C} \). Substitute \( B = 4 \), \( A = 1 \), and \( C = -2 \): \( \tan(2\theta) = \frac{4}{1 - (-2)} = \frac{4}{3} \). Thus, \( 2\theta = \tan^{-1}\left(\frac{4}{3}\right) \). Solve for \( \theta \).
03

Apply rotation

Using \( \cos(\theta) \) and \( \sin(\theta) \), compute the new variables \( x' \) and \( y' \) from \( x \) and \( y \) with the rotation formulas: \( x = x'\cos(\theta) - y'\sin(\theta) \) and \( y = x'\sin(\theta) + y'\cos(\theta) \). Substitute back into the original equation.
04

Simplify the rotated equation

After substitution and simplification, collect terms to remove the \( x'y' \)-term, which is possible after rotation. The new equation represents a conic in its standard form, free of the cross product term.
05

Identify the conic section

Analyze the simplified equation's coefficients: if \( A'C' + B'^2 = 0 \), it's a parabola; if \( A'C' > 0 \), it's an ellipse; if \( A'C' < 0 \), it's a hyperbola. Apply this to identify the conic section.
06

Sketch the graph

Use the identified conic type and its properties to sketch a graph. Indicate the axes after rotation as the new coordinate system \( x'y' \), ensuring it represents the transformed conic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are fascinating geometric shapes, each derived from intersecting a plane with a cone at different angles and positions. The primary types are circles, ellipses, parabolas, and hyperbolas.
  • A circle is formed when the intersecting plane is perpendicular to the cone's axis.
  • An ellipse occurs when the plane intersects the cone at an angle but does not pass through its base.
  • A parabola is formed when the plane is parallel to a slant of the cone.
  • A hyperbola emerges when the plane intersects both nappes (the two distinct pieces) of the cone.
Conic sections appear not only in mathematics but also in various fields like physics and engineering. Recognizing these forms within equations and sketches can help identify and understand these natural structures.
Equation Transformation
Equation transformation is a mathematical technique used to manipulate a given equation into a desired form. It often involves operations like rotation, translation, and scaling.

In the context of conic sections, rotation is particularly useful for removing the cross-term, such as the \( xy \)-term in an equation of a conic. To perform a rotation:
  • Determine the angle \( \theta \) using \( \tan(2\theta) = \frac{B}{A - C} \), which helps align the equation with the axes.
  • Once \( \theta \) is found, apply trigonometric formulas to rotate the coordinate axes, eliminating the cross-term.
This simplification makes it easier to identify the type of conic section present in the equation. Equation transformation is an essential skill that enhances problem-solving and equation manipulation, aiming to simplify and make equations more interpretable.
Graph Sketching
Graph sketching is an important skill that allows one to visualize mathematical functions and equations.
When dealing with conic sections, sketching becomes a crucial step after identifying the equation's type. Here are some tips to create accurate sketches:
  • Identify the conic section type: circle, ellipse, parabola, or hyperbola. This determines the general shape.
  • Use the equation's coefficients to find key features like the center, vertices, and asymptotes (for hyperbolas).
  • Mark the axes of symmetry, which are often aligned with significant features of the conic.
  • For rotated conics, adjust the axes to match the new coordinate system after transformation.
By accurately sketching the graph, you visually represent the conic's behavior and symmetry, enabling better understanding and analysis of its properties.

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Most popular questions from this chapter

The prolate cycloid $$ x=2-\pi \cos t, \quad y=2 t-\pi \sin t \quad(-\pi \leq t \leq \pi) $$ crosses itself at a point on the \(x\) -axis (see the accompanying figure). Find equations for the two tangent lines at that point.

Find the eccentricity and the distance from the pole to the directrix, and sketch the graph in polar coordinates. (a) \(r=\frac{4}{2+3 \cos \theta}\) (b) \(r=\frac{5}{3+3 \sin \theta}\)

Use the following values, where needed: radius of the Earth \(=4000 \mathrm{mi}=6440 \mathrm{~km}\) year (Earth year) \(=365\) days (Earth days) \(1 \mathrm{AU}=92.9 \times 10^{6} \mathrm{mi}=150 \times 10^{6} \mathrm{~km}\) (a) Let \(a\) be the semimajor axis of a planet's orbit around the Sun, and let \(T\) be its period. Show that if \(T\) is measured in days and \(a\) is measured in kilometers, then \(T=\left(365 \times 10^{-9}\right)(a / 150)^{3 / 2}\) (b) Use the result in part (a) to find the period of the planet Mercury in days, given that its semimajor axis is \(a=57.95 \times 10^{6} \mathrm{~km}\) (c) Choose a polar coordinate system with the Sun at the pole, and find an equation for the orbit of Mercury in that coordinate system given that the eccentricity of the orbit is \(e=0.206\). (d) Use a graphing utility to generate the orbit of Mercury from the equation obtained in part (c).

What are some of the advantages of expressing a curve parametrically rather than in the form \(y=f(x)\) ?

Prove: If \(B \neq 0\), then the graph of \(x^{2}+B x y+F=0\) is a hyperbola if \(F \neq 0\) and two intersecting lines if \(F=0 .\)
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