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:

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

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