How do you find the equation of an ellipse with major axis vertical and passes through points (0,4) and (2,0)?

1 Answer
Dec 28, 2016

This problem can only be done, if one assumes that two given points, #(0,4) and (2,0)#, are the upper vertex on major axis and right vertex on the minor axis, respectively.

Explanation:

The general Cartesian equation for an ellipse with a vertical major axis is:

#(y - k)^2/a^2 + (x - h)^2/b^2 = 1" [1]"#

Where x and y correspond any point, #(x, y)# in the ellipse, h and k correspond to the center point #(h, k)#, "a" is the length of the semi-major axis, and "b" is the length of the semi-minor axis.

Using the previously stated assumption.

The upper vertex on the major axis will have the coordinates:

#(h, k + a)#

The right vertex on the minor axis will have the coordinates:

#(h + b, k)#

Matching these two point to the given points:

#(h, k + a) = (0,4)#
#(h + b, k) = (2,0)#

This allows us to write the following equations:

#h = 0, k + a = 4, h + b = 2, and k = 0#

Substituting 0 for h and k:

#h = 0, a = 4, b = 2, and k = 0#

Substitute these 4 values into equation [1]:

#(y - 0)^2/4^2 + (x - 0)^2/2^2 = 1" [2]"#

Equation [2] is the answer.

Here is a graph with the ellipse and the two points plotted:

enter image source here