The equation of an ellipse having a major axis of length 18 and foci located at (4,7) and (4,11) is given by: Option A: (x-4)^2/77 + (x-9)^2/81 = 1
Suppose that the major axis is of the length 2a units, and that minor axis is of 2b units, and let the ellipse is centered on (h,k) with major ellipse parallel to x-axis, then the equation of that ellipse would be:
[tex]\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} =1[/tex]
Coordinates of its foci would be: [tex](h \pm c, k)[/tex] where [tex]c^2 = a^2 - b^2[/tex]
If its major axis is parallel to y-axis, then,
coordinates of its foci would be: [tex](h , k\pm c)[/tex] where [tex]c^2 = a^2 - b^2[/tex] and its equation would be:
[tex]\dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} =1[/tex]
For this case, we're specified that:
Assume that:
Then, we get:
2a = 18, or a = 9 units
Also, as we have:
Coordinates of its foci as: (4,7), and (4,11)
Foci lie on major axis, and the x-coordinate of both foci is constant, so major axis is parallel to y-axis.
For such ellipse, its foci is [tex](h , k\pm c)[/tex]
The value of c is always non-negative as a ≥ b always as 2a is length of major axis and 2b is length of minor axis.
Thus, we get:
[tex]k - c = 7\\k + c = 11[/tex] (c isn't 0 as it has some effects as foci' y coordinates are different. Thus, c > 0. Andt therefore a fixed number - positive number < that fixed number + that positive number, therefore we alloted 7 and 11 to those foci).
From those 2 equations, we get:
[tex]k = c + 7\\k+ c =c+7+c = 11\\c = 2\\k = 2+7 = 9\\[/tex]
Also, from foci coordinates, we get h = 4
Thus, center of the ellipse considered is (h,k) = (4,9)
Also, as we've got [tex]c^2 = a^2 - b^2[/tex] and c = 2, and a = 9, therefore, we get:
[tex]2^2 =9^2 - b^2\\b^2 = 77[/tex]
Thus, the equation of the ellipse considered is:
[tex]\dfrac{(x-h)^2}{b^2} + \dfrac{(y-k)^2}{a^2} =1\\\\\dfrac{(x-4)^2}{77} + \dfrac{(y-9)^2}{81} =1[/tex]
Thus, the equation of an ellipse having a major axis of length 18 and foci located at (4,7) and (4,11) is given by: Option A: (x-4)^2/77 + (x-9)^2/81 = 1
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