/*! 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 62 Find the points of intersection ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 62

Find the points of intersection of \(x^{2}+4 y^{2}=20\) and \(x+2 y=6\)

Short Answer

Expert verified
The intersection points are (4, 1) and (2, 2).

Step by step solution

01

Identify Equations

The problem provides two equations. The first equation is a conic section: \(x^2 + 4y^2 = 20\), which is an ellipse. The second equation is linear: \(x + 2y = 6\). Our task is to find the points where these two curves intersect.
02

Solve the Linear Equation for x

Start by isolating \(x\) in the second equation \(x + 2y = 6\). Solve for \(x\) to express it in terms of \(y\):\[x = 6 - 2y\].
03

Substitute into the Quadratic Equation

Replace \(x\) in the first equation \(x^2 + 4y^2 = 20\) with the expression found in Step 2, \(6 - 2y\). This gives us:\[(6 - 2y)^2 + 4y^2 = 20\].
04

Expand and Simplify

Expand \((6 - 2y)^2\) to get \(36 - 24y + 4y^2\). Substitute this back into the equation to get:\[36 - 24y + 4y^2 + 4y^2 = 20\]. Combine like terms to obtain:\[8y^2 - 24y + 36 = 20\].
05

Rearrange into a Standard Quadratic Equation

Subtract 20 from both sides of the equation to get:\[8y^2 - 24y + 16 = 0\]. Divide the entire equation by 8 to simplify:\[y^2 - 3y + 2 = 0\].
06

Factor the Quadratic

The equation \(y^2 - 3y + 2 = 0\) can be factored into:\[(y - 1)(y - 2) = 0\]. This gives two potential solutions for \(y\), which are \(y = 1\) and \(y = 2\).
07

Find Corresponding x-values

Use the expression \(x = 6 - 2y\) to find the corresponding \(x\) values for each \(y\).For \(y = 1\), \(x = 6 - 2(1) = 4\).For \(y = 2\), \(x = 6 - 2(2) = 2\).
08

Intersection Points

The two points of intersection are \((4, 1)\) and \((2, 2)\).

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.

Ellipse Equations
An ellipse is a type of conic section, which is a curve obtained by intersecting a cone with a plane. The standard form of an ellipse equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\], where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively. When these axes are equally distributed around the origin, they define the size and shape of the ellipse.
  • The equation given in the exercise is \(x^2 + 4y^2 = 20\). By dividing everything by 20, it can be rewritten as \(\frac{x^2}{20} + \frac{y^2}{5} = 1\).
  • This indicates an ellipse with a semi-major axis of \(\sqrt{20}\) and a semi-minor axis of \(\sqrt{5}\). The longer axis is along the x-direction, indicating the ellipse is wider than it is tall.
Ellipses often appear in physics and astronomy, for example, describing orbits of planets or satellites.
Linear Equations
Linear equations represent straight lines. They can be expressed in many forms, with the most common being the slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.In this exercise, the linear equation \(x + 2y = 6\) is given. It provides a direct, simple relationship between \(x\) and \(y\). This can also be rearranged to the form \(y = -\frac{1}{2}x + 3\), showing:
  • A slope of -\(\frac{1}{2}\), indicating that for every 1 unit increase in \(x\), \(y\) decreases by \(0.5\).
  • The y-intercept is 3, meaning the line crosses the y-axis at (0,3).
This straight line intersects the ellipse at two points, requiring the manipulation and substitution as outlined in the solution to solve for the intersection points.
Quadratic Factoring
Quadratic factoring involves expressing a quadratic equation as a product of its factors. For a quadratic in the form \(ax^2 + bx + c = 0\), factoring can simplify solving. In this context, the equation \(y^2 - 3y + 2 = 0\) was obtained after rearranging and simplifying from the intersecting ellipse and line equations. The factors are:
  • \((y - 1)(y - 2) = 0\), indicating solutions \(y = 1\) and \(y = 2\).
  • These factors provide potential values for \(y\) that satisfy both initial equations simultaneously.
Factoring simplifies the process of finding solutions by breaking down the equation into more workable parts, making it easier to solve by inspection or using the zero product property.
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include shapes such as circles, ellipses, parabolas, and hyperbolas. Each conic section has a distinct set of properties and equations that describe their curves.
  • Circles are a special type of ellipse with equal semi-major and semi-minor axes.
  • Ellipses result when the intersection of the plane and the cone is such that it completely passes through one nap but not the other.
  • Parabolas are formed when the plane is parallel to one of the generating lines of the cone.
  • Hyperbolas result when the plane intersects both naps.
Understanding conic sections is crucial in a wide range of applications, from engineering to physics, where they help describe system behaviors and optimize designs.

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

In order to graph a polar equation such as \(r=f(t)\) using a parametric equation grapher, you must replace this equation by \(x=f(t) \cos t\) and \(y=f(t) \sin t .\) These equations can be obtained by multiplying \(r=f(t)\) by \(\cos t\) and \(\sin t\), respectively. Confirm the discussions of conics in the text by graphing \(r=4 e /(1+e \cos t)\) for \(e=0.1,0.5,0.9,1,1.1\) and \(1.3\) on \([-\pi, \pi]\).

Eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form. Finally, graph the equation showing the rotated axes. 3 x^{2}+10 x y+3 y^{2}+10=0

Sketch the given curves and find their points of intersection. $$ r=1-\cos \theta, r=1+\cos \theta $$

Show that, if \(A+C\) and \(\Delta=4 A C-B^{2}\) are both positive, then the graph of \(A x^{2}+B x y+C y^{2}=1\) is an ellipse (or circle) with area \(2 \pi / \sqrt{\Delta}\). (Recall from Problem 55 of Section \(10.2\) that the area of the ellipse \(x^{2} / p^{2}+y^{2} / q^{2}=1\) is \(\left.\pi p q .\right)\)

. Using a computer algebra system, plot the following parametric curves for \(0 \leq t \leq 2 .\) Describe the shape of the curve in each case and the similarities and differences among all the curves. (a) \(x=t, y=t^{2}\) (b) \(x=t^{3}, y=t^{6}\) (c) \(x=-t^{4}, y=-t^{8}\) (d) \(x=t^{5}, y=t^{10}\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.