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.
The graph of the equation and its characteristics.
The given equation is,
The basic equation of the ellipse is,
The center of ellipse is given by,
The length of major axis is .
The length of minor axis is .
The focal length is given by,
The co-ordinates of foci is given by,
Consider the given equation,
Now complete the square as shown below,
Compare the above equation with
This is the ellipse with center .
The vertices are , and co vertices are .
The end points of major and minor axes of the un-shifted ellipse are,
The end points of major axes are,
The end points of minor axes are,
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