Complete the square to determine whether the equation represents 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 hyperbola, find the center, foci, vertices and asymptotes. Sketch the graph of the equation. If the equation has no graph, explain why.
x2+4y2+20x−40y+300=0Subtract 300x2+4y2+20x−40y=−300Factor and group terms(x2+20x+)+4(y2−10y+)=−300Complete the square: add (202)2=100 or both sides and (−102)2=25 on the left and 100 on the right(x2+20x+100)+4(y2−10y+25)=−300+100+100Perfect square(x+10)2+4(y−5)2=−100Divide both sides by −100−(x+10)2100−(y−5)2100=1
We can see that the equation has no solution since the sum of the squares can never be a negative value. Thus, the equation is a degenerate conic and it has no graph.
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