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 hyperbola with a horizontal transverse axis which is used is as follows,
Where and are constant values.
Expression to find vertices is,
Expression to find Foci is,
Expression to find Asymptotes is,
Consider the given equation,
Now complete the square as shown below,
The above equation is an equation of a hyperbola with center and horizontal transverse axis.
The shape of this graph is same as that of the un-shifted hyperbola has having equation as,
Hyperbola with center at origin,
Thus the un-shifted ellipse foci are,
Thus, the shifted hyperbola foci are,
Therefore, the foci of the shifted hyperbola are and
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