How do you write an equation of an ellipse given endpoints of major axis at (-11,5) and (7,5) and endpoints of the minor axis at (-2,9) and (-2,1)?

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Answer 1 · 2016-12-12T22:50:04

Please see the explanation for steps leading to the equation.

The endpoints, #(-11, 5) and (7,5)#, of the major axis (where the x coordinate changes) have a general form of:

#(h - a, k) and (h + a, k)#

This allows us to write the following equations:

#"[1] "##k = 5#
#"[2] "##h - a = -11#
#"[3] "##h + a = 7#

The endpoints, #(-2, 1) and (-2,9)#, of the minor axis (where the y coordinate changes) have a general form of:

#(h, k - b) and (h, k + b)#

This allows us to write the following equations:

#"[4] "##h = -2#
#"[5] "##k - b = 1#
#"[6] "##k + b = 9#

Subtracting equation [2] from [3] gives us:

#2a = 18#

#a = 9#

Subtracting equation [5] from [6] gives us:

#2b = 8#

#b = 4#

All that remains, is to substitute these values into the general form for an ellipse with a horizontal major axis:

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

#(x - -2)^2/9^2 + (y - 5)^2/4^2 = 1#