Monday, July 29, 2019

College Algebra, Chapter 8, 8.4, Section 8.4, Problem 34

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+20x40y+300=0Subtract 300x2+4y2+20x40y=300Factor and group terms(x2+20x+)+4(y210y+)=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(y210y+25)=300+100+100Perfect square(x+10)2+4(y5)2=100Divide both sides by 100(x+10)2100(y5)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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