Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

1st Edition

ISBN: 9780078682278

Author: McGraw-Hill, Berchie Holliday

Publisher: Glencoe/McGraw-Hill

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Concept explainers

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

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Question

Chapter 10.7, Problem 33E

To determine

Tofind:the graph of the equation and the graph of equation is degenerated case.

Expert Solution & Answer

Answer to Problem 33E

The new equation so form is y=−2x−3 .

Explanation of Solution

Given:

(x−2)2−(x+3)2=5(y+2) .

Concept used:

The general equation for conic section is:

Ax2+Bxy+Cy2+Dx+Ey+F=0.

Where A,B,C,D,E and F are constant.

As by changing the values of some of the constants, the shape of the corresponding conic also changes.

If B2−4AC , is less than zero, if a conic exists, it will be either a circle or an ellipse.

If B2−4AC ,equals to zero, if a conic exists, it will be parabola.

If B2−4AC id greater than zero, if a conic exists, it will be hyperbola.

Calculation:

Consider the equation of the conic:

(x−2)2−(x+3)2=5(y+2) .

Rearrange the equation in the general form of conic:

Ax2+Bxy+Cy2+Dx+Ey+F=0.

B=0 if conic parallel to co-ordinate axis else B≠0 .

(x2−2.2.x+4)−(x2+2.3.x+9)=5(y+2).⇒x2−4x+4−x2−6x−9=5(y+2).⇒−10x−5=5(y+2).−2x−y−3=0.

Comparing the equation with general form of conic above:

A=0,B=0,C=0,D=−2,E=−1,F=−3.

Now,

B2−4AC=0−4.0.0=0.

As in the given equation (x−2)2−(x+3)2=5(y+2) value of B2−4AC=0 so, at first glance the given equation may appear to be that of a parabola but a closer inspection reveals that this is a degenerate case.

So, by solving the equation (x−2)2−(x+3)2=5(y+2) to get:

−2x−y−3=0.y=−2x−3.

Now find the certain points to graph the straight line y=−2x−3 :

The graph of equation y=−2x−3 is:

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 10.7, Problem 33E

Hence, the new equation so form is y=−2x−3 .

Chapter 10.7, Problem 32E

Chapter 10.7, Problem 34E

Chapter 10 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

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Finding The Focus and Directrix of a Parabola - Conic Sections; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=KYgmOTLbuqE;License: Standard YouTube License, CC-BY

Tofind: the graph of the equation and the graph of equation is degenerated case.

Concept explainers

Videos

Tofind:the graph of the equation and the graph of equation is degenerated case.

Answer to Problem 33E

Explanation of Solution

Chapter 10 Solutions

Additional Math Textbook Solutions