/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 (a) Use the identification theor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 5

(a) Use the identification theorem (12.14) to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. (b) Use a suitable rotation of axes to find an equation for the granh in an \(x^{\prime} y^{\prime}\) -plane, and sketch the graph, labeling vertices. $$ 11 x^{2}+10 \sqrt{3} x y+y^{2}=4 $$

Short Answer

Expert verified
The graph is a hyperbola, with vertices located along the transformed axes after rotation.

Step by step solution

01

Identify The Quadratic Form

The given equation is \( 11x^2 + 10\sqrt{3}xy + y^2 = 4 \). Recognize it fits the general quadratic form \( Ax^2 + Bxy + Cy^2 = D \) with \( A = 11 \), \( B = 10\sqrt{3} \), and \( C = 1 \). This step prepares for classification using the identification theorem.
02

Calculate Discriminant for Classification

The discriminant \( \Delta \) is calculated as \( B^2 - 4AC \). Substitute \( B = 10\sqrt{3} \), \( A = 11 \), and \( C = 1 \) to find \( \Delta = (10\sqrt{3})^2 - 4 \times 11 \times 1 = 300 - 44 = 256 \).
03

Classify the Conic Section

According to the identification theorem, the conic section is classified based on the discriminant:\ \( \Delta > 0 \) indicates a hyperbola;\ \( \Delta = 0 \) is a parabola;\ \( \Delta < 0 \) is an ellipse.\ Here \( \Delta = 256 > 0 \), so the graph is a hyperbola.
04

Determine Rotation Angle

To eliminate the \( xy \)-term by rotating the axes, compute the rotation angle \( \theta \) using \( \cot(2\theta) = \frac{A - C}{B} = \frac{11 - 1}{10\sqrt{3}} = \frac{10}{10\sqrt{3}} = \frac{1}{\sqrt{3}} \). Solidify understanding that \( \theta = 30^\circ \) or \( \theta = -60^\circ \).
05

Rotate Axes to Simplify Equation

Apply the rotation formulas \( x = x' \cos \theta - y' \sin \theta \) and \( y = x' \sin \theta + y' \cos \theta \). For \( \theta = 30^\circ \), \( \cos\theta = \sqrt{3}/2 \) and \( \sin\theta = 1/2 \). Substitute and simplify to find equation in \( x' \) and \( y' \) form.
06

Simplify and Sketch the Graph

After substitution, the equation simplifies to \( 8x'^2 - 6y'^2 = 4 \). Dividing through by 4 gives \( 2x'^2 - \frac{3}{2}y'^2 = 1 \), which is in the form \( \frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1 \). Sketch a hyperbola with vertices focusing on symmetry axes, centered at the origin, using these parameters.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Conic Sections
Conic sections are the curves obtained by intersecting a double-napped cone with a plane at various angles. These intersections create the following curves:
  • Circle: The plane cuts the cone parallel to its base.
  • Ellipse: The plane intersects the cone at an angle, forming an elongated circle.
  • Parabola: If the plane is parallel to a slant edge of the cone, we get a parabola.
  • Hyperbola: Occurs when the plane cuts through both nappes of the cone, resulting in two opposite curves.
Conic sections are described by quadratic equations in two variables, typically written as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). They have significant implications in mathematics and science, providing models for planetary orbits, light reflection properties, and more.
Discriminant
The discriminant in a quadratic form equation helps us determine the nature of a conic section without directly graphing it. For the general quadratic equation \(Ax^2 + Bxy + Cy^2 = D\), the discriminant \(\Delta\) is calculated as \(\Delta = B^2 - 4AC\).
Here is how we interpret the discriminant value:
  • If \(\Delta > 0\), the conic is a hyperbola. This suggests the plane intersects both nappes of the cone.
  • If \(\Delta = 0\), it represents a parabola where the plane is tangent to the cone.
  • If \(\Delta < 0\), it indicates an ellipse, which describes the plane's slight angle intersection with the cone.
In the provided exercise, we calculated \(\Delta = 256\). With \(\Delta > 0\), we confirm that the conic section is a hyperbola.
Rotation of Axes
The rotation of axes is a powerful method to simplify the quadratic equation by eliminating the \(xy\)-term. This is particularly useful for analyzing conics that are rotated from the standard position.
To achieve this simplification, we find the rotation angle \(\theta\) using the formula:
  • \(\cot(2\theta) = \frac{A - C}{B}\)
In our example, substituting \(A = 11\), \(C = 1\), and \(B = 10\sqrt{3}\), we find that \(\theta\) corresponds to either \(30^\circ\) or \(-60^\circ\).
The transformation formulas are:
  • \(x = x' \cos\theta - y' \sin\theta\)
  • \(y = x' \sin\theta + y' \cos\theta\)
These rotated coordinates \((x', y')\) allow for a simplified equation that is easier to analyze and graph.
Hyperbola
A hyperbola is a type of conic section characterized by two open, mirror-image curves. In contrast to ellipses and parabolas, hyperbolas possess unique properties:
  • They have two distinct branches, which expand outward but never close like an ellipse.
  • Each branch has a vertex, and these form the principal axis of the hyperbola.
  • Hyperbolas have asymptotes, indicating the direction in which each branch of the curve extends.
The standard form of a hyperbola is \(\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1\) or \(\frac{y'^2}{b^2} - \frac{x'^2}{a^2} = 1\).
In the example, the simplified form after rotation becomes \(2x'^2 - \frac{3}{2}y'^2 = 1\), clearly showing it's a hyperbola. The parameters \(a\) and \(b\) define the hyperbola's extent along its axes and how rapidly the branches open.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 2, \quad y\) -intercepts \(\pm_{3}^{1}\)

A searchlight reflector is designed so that every cross section containing its axis of symmetry is a parabola with the light source at the focus. Where is the focus if the reflector is 3 feet across at the opening and 1 foot deep?

Prove that the point on a parabola that is closest to the focus is the vertex.

Find an equation of the parabola that satisfies the given conditions. vertex at the origin, symmetric to the \(y\) -axis, and passing through the point (2,-3)

Let \(R\) be the region bounded by the parabola \(x^{2}=4 y\) and the line \(/\) through the focus that is perpendicular to the axis of the parabola. (a) Find the area of \(R\). (b) If \(R\) is revolved about the \(y\) -axis, find the volume of the resulting solid. (c) If \(R\) is revolved about the \(x\) -axis, find the volume of the resulting solid.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.