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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

Degenerate Conic What value of F must be if the graph of the equation

4 x 2 + y 2 + 4 ( x 2 y ) + F = 0 is ( a ) . an ellipse, ( b ) . a single point ( c ) . the empty set

To determine

(a)

The value of F for which the graph of the equation 4x2+y2+4(x2y)+F=0 is an ellipse.

Explanation

Given:

The given equation is

4x2+y2+4(x2y)+F=0

Approach:

Take t=a+b, u=ab and w=abcaf2bg2 … (1)

Then the equation is an ellipse if tw<0

Calculation:

Compare the equation with the general second-degree equation of an ellipse ax2+by2+2gx+2fy+c=0

Then, a=4, b=1, g=2, f=4 and c=F.

Take tw<0

From equation (1)

(a+b)(abcaf2bg2)<0

Substitute the values, a=4, b=1, g=2, f=4 and c=F

To determine

(b)

The value of F for which the graph of the equation 4x2+y2+4(x2y)+F=0 is a single point.

To determine

(c)

The value of F has to be greater than 17 for the given equation 4x2+y2+4(x2y)+f=0 to represents an empty set.

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