How do you graph the ellipse #(x-4)^2/8+(y-2)^2/18=1# and find the center, the major and minor axis, vertices, foci and eccentricity?

1 Answer
KillerBunny
May 2, 2018

Center: #4,2#
Major axis: length #3sqrt(2)#, vertical
Minor axis: legnth #2sqrt(2)#, horizontal
Vertices: #(2sqrt(2),0),(-2sqrt(2),0),(0,3sqrt(2)),(0,-3sqrt(2))#
Foci: #(0,-sqrt(10)),(0,sqrt(10))#
Eccentricity: #sqrt(5)/3#

Explanation:

Given the general formula

#(x-x_0)^2/(a^2) + (y-y_0)^2/(b^2) = 1#

you have that:
- #x_0# is the #x# component of the center
- #y_0# is the #y# component of the center
- #a# is the horizontal semi-axis
- #b# is the vertical semi-axis

This easily answers the question about center, axes and vertices.

For the rest, you have a formula for everything:

  • Foci: #c = sqrt(b^2-a^2)# (in general, major axis minus minor axis)
  • Eccentricity = #c/b# (in general, focus divided by major axis)
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