SOLUTION: Name the coordinates of the four vertices, the two foci, and the equations of the major and minor axes for the ellipse 25x^2+4y^2-150x+32y+189=0

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Name the coordinates of the four vertices, the two foci, and the equations of the major and minor axes for the ellipse 25x^2+4y^2-150x+32y+189=0      Log On


   

Question 817915: Name the coordinates of the four vertices, the two foci, and the equations of the major and minor axes for the ellipse 25x^2+4y^2-150x+32y+189=0
Answer by lwsshak3(11628) About Me  (Show Source):
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Name the coordinates of the four vertices, the two foci, and the equations of the major and minor axes for the ellipse
25x^2+4y^2-150x+32y+189=0
rearrange terms:
25x^2-150x+4y^2+32y=-189
complete the square:
25(x^2-6x+9)+4(y^2+8y+16)=-189+225+64
25(x-3)^2+4(y+4)^2=100
%28x-3%29%5E2%2F4%2B%28y%2B4%29%5E2%2F25=1
This is an equation of an ellipse with vertical major axis
Its standard form of equation:
%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1,a>b, (h,k)=(x,y) coordinates of center
..
For given ellipse:
center: (3,-4)
a^2=25
a=√25=5
vertices:(3,-4±a)=(3,-4±5)=(3,-9) and (3,1)(ends of major axis)
b^2=4
b=2
vertices:(3±b,-4)=(3±2,-4)=(1,-4) and (5,-4)(ends of minor axis)
c^2=a^2-b^2=25-4=21
c=√21≈4.6
foci:(3,-4±c)=(3,-4±4.6)=(3,-8.6) and (3,.6)