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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Chapter 10.7, Problem 34E
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 34E

  2(x−(−1))2+6(y−1)2=0 .

Explanation of Solution

Given:

  2x2+6y2+8x−12y+14=0.

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:

  2x2+6y2+8x−12y+14=0.

Rearrange the equation in the general form of conic:

  Ax2+Bxy+Cy2+Dx+Ey+F=0.

Comparing the equation with general form of conic above:

  A=2,B=0,C=6,D=8,E=−12,F=14.

Now,

  B2−4AC=02−4.2.6=−48.

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

  2x2+6y2+8x−12y+14=0. represents an ellipse.

So, write the equation 2x2+6y2+8x−12y+14=0 in the standard form,

  2(x2+4x+4)+6(y2−2y+1)=0.2(x−(−1))2+6(y−1)2=0....(i)

The standard equation of an ellipse is:

  (x−h)2a2+(y−k)2b2=1.....(ii)

Where (h,k) : center, a: semi major axis,b: semi minor axis.

Compare the equation (i) and (ii) :

  h=−1,k=1,a=0,b=0.

Therefore, the equation 1 describe a degenerate ellipse or point with coordinate (−1,1) .

The graph of the equation 2x2+6y2+8x−12y+14=0 is:

  Advanced Mathematical Concepts: Precalculus with Applications, Student Edition, Chapter 10.7, Problem 34E

Hence, the new equation so form is 2(x−(−1))2+6(y−1)2=0 .

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Chapter 10.7, Problem 33E
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Chapter 10.7, Problem 35E

Chapter 10 Solutions

Advanced Mathematical Concepts: Precalculus with Applications, Student Edition

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