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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

SKILLS

47-58 ■ Graphing Shifted Conics Complete the square to deter-mine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyper-bola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why.

9 x 2 36 x + 4 y 2 = 0

To determine

The graph of the equation and its characteristics.

Explanation

Given:

The given equation is,

9x236x+4y2=0

Approach:

The basic equation of the ellipse is,

(xh)2a2+(yk)2b2=1

The center of ellipse is given by,

(h,k)

The length of major axis is 2b.

The length of minor axis is 2a.

The focal length is given by,

c2=b2a2

The co-ordinates of foci is given by,

(h±c,k)

Calculation:

Consider the given equation,

9x236x+4y2=0

Now complete the square as shown below,

9x236x+4y2=09x236x+36+4y236=09(x2)2+4y2=36(x2)24+y29=1

Compare the above equation with (xh)2a2+(yk)2b2=1

This is the ellipse with center (2,0).

And,

a2=4a=±2b2=9b=±3

The vertices are (±2,0), and co vertices are (0,±3).

The end points of major and minor axes of the un-shifted ellipse are,

The end points of major axes are,

(h+a,k)=(2+2,0)=(4,0)(h+a,k)=(22,0)=(0,0)

The end points of minor axes are,

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