An elliptical stained-glass insert is to be fitted in a 4ft by 6ft rectangular opening (see figure). Using the coordinate system shown, find an equation for the ellipse.

Question

An elliptical stained-glass insert is to be fitted in a 4ft by 6ft rectangular opening (see figure).  Using the coordinate system shown, find an equation for the ellipse. ​

Quizplus · Accepted Answer

We can find the center of the ellipse by taking the average of the x- and y-coordinates of the vertices:
Center = ((-1+5)/2,(-1+7)/2) = (2,3)
Then, we can use the distance formula to find the distance between the center and one of the vertices:
distance = sqrt((5-2)^2 + (-1-3)^2) = sqrt(45)
The distance between the center and one of the co-vertices (the endpoints of the minor axis) is half of this distance, so:
co-vertex distance = sqrt(45)/2
Since the length of the rectangle is 6ft, the distance between the center and one of the endpoints of the major axis is also 3ft.
Therefore, the equation of the ellipse is:
(x-2)^2/3^2 + (y-3)^2/(sqrt(45)/2)^2 = 1
Simplifying and rationalizing the denominator gives us:
(x-2)^2/9 + 4(y-3)^2/45 = 1
Multiplying both sides by 45 gives us the standard form:
5(x-2)^2 + 36(y-3)^2 = 180

An Elliptical Stained-Glass Insert Is to Be Fitted in a 4ft