How do you find an equation for the ellipse with vertices at (-6,4) and (10,4); focus at (8,4)?

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Answer 1 · 2016-11-16T01:12:12

Please see the explanation.

The equation of an ellipse:

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

Has vertices at #(h +-a, k)#
Has foci at #(h +-sqrt(a^2 -b^2), k)#

Use the vertices to write 3 equations:

#k = 4" [1]"#
#h - a = -6" [2]"#
#h + a = 10" [3]"#

Use equations [2] and [3] to solve for h and a:

#2h = 4#
#h = 2#
#a = 8#

Use the focus to write another equation:

#8 =h + sqrt(a^2 - b^2)#

Substitute values for h and a:

#8 = 2 + sqrt(8^2 - b^2)#

#6 = sqrt(64 - b^2)#

#36 = 64 - b^2#

#b^2 = 64 - 36#

#b^2 = 28#

#b = sqrt(28)#

Substitute the values into the standard form:

#(x - 2)^2/8^2 + (y - 4)^2/(sqrt(28))^2 = 1#