Wednesday, November 18, 2015

College Algebra, Chapter 8, 8.4, Section 8.4, Problem 24

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.


9x236x+4y2=0Factor and group terms9(x24x+)+4y2=0Complete the square: add (42)2=4 on the right side and 36 on the left side9(x24x+4)+4y2=36Complete the square9(x2)2+4y2=36Divide by 36(x2)24+y29=1


We can say that the equation is an ellipse since it is the sum of the squares. The equation (xh)2b2+(yk)2a2=1 has a vertical major axis and vertices on (0,±a), center on (h,k) and foci (0,±c) where c=a2b2.

Also, the length of its major and minor axis is 2a and 2b respectively, So if a2=9 and b2=4, then a=3,b=2 and c=94=5.

Thus, by applying transformations,

center (2,0)

vertices (2,0±3)(2,±3)

foci (2,0±5)(2,±5)

length of major axis 6

length of minor axis 4

Therefore, the graphs is

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