Advanced Mathematical Concepts: Precalculus with Applications, Student Edition
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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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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Chapter 10.7, Problem 36E
To determine

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

Expert Solution & Answer
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Answer to Problem 36E

  x2+y2=0 .

Explanation of Solution

Given:

  (x−2)2+(y−2)2+4(x+y)=8.

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+(y−2)2+4(x+y)=8.

The general equation in the form of conic:

  Ax2+Bxy+Cy2+Dx+Ey+F=0....(i)

Rearrange the equation in this form as:

  x2−4x+4+y2−4y+4+4x+4y=8.⇒x2+y2=0......(ii)

Comparing equation (i) and (ii) :

  A=1,B=0,C=1,D=0,E=0,F=0.

Now,

  B2−4AC=02−4.1.1=−4.

While B2−4AC<0 and A=C, from the table of comparison of conic, it follows that

  (x−2)2+(y−2)2+4(x+y)=8. represents the circle.

So, write the equation x2+y2=0 in the standard form as:

  (x−0)2+(y−0)2=02....(iii)

The standard equation of an ellipse is:

  (x−h)2+(y−k)2=r2....(iv)

Where (h,k) : center, r:radius .

Compare the equation (iii) and (iv) :

  h=0,k=0,r=0.

Therefore, the equation (iii) describe a point circle with the center (0,0) .

The graph of the equation (x−2)2+(y−2)2+4(x+y)=8 is:

  Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 10.7, Problem 36E

Hence, the new equation so form is x2+y2=0 .

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Chapter 10.7, Problem 35E
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Chapter 10.7, Problem 37E

Chapter 10 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

Ch. 10.1 - Prob. 1CFUCh. 10.1 - Prob. 2CFUCh. 10.1 - Prob. 3CFUCh. 10.1 - Prob. 4CFUCh. 10.1 - Prob. 5CFUCh. 10.1 - Prob. 6CFUCh. 10.1 - Prob. 7CFUCh. 10.1 - Prob. 8CFUCh. 10.1 - Prob. 9CFUCh. 10.1 - Prob. 10CFU
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