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.
9x2−36x+4y2=0Factor and group terms9(x2−4x+)+4y2=0Complete the square: add (42)2=4 on the right side and 36 on the left side9(x2−4x+4)+4y2=36Complete the square9(x−2)2+4y2=36Divide by 36(x−2)24+y29=1
We can say that the equation is an ellipse since it is the sum of the squares. The equation (x−h)2b2+(y−k)2a2=1 has a vertical major axis and vertices on (0,±a), center on (h,k) and foci (0,±c) where c=√a2−b2.
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=√9−4=√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
No comments:
Post a Comment