47-58 Graphing Shifted Conics Complete the square to determine 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 the lengths of the major and minor axes. If it is parabola, find the vertex, focus, and directrix. If it is hyperbola, 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 for an ellipse with a vertical major axis which is used is as follows,
Where and are constant values.
Expression to find vertices is,
Expression to find Foci is,
Consider the given equation,
Now complete the square as shown below,
Further solve the above equation.
This is an equation of a shifted ellipse with center .
It is obtained from the equation of the ellipse at the center,
The end points of the minor and major axes of the un-shifted ellipse are,
The end points of the minor and major axes of the shifted ellipse are,
Thus the un-shifted ellipse foci are,
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