find the equation of an ellipse iwth center at the origina, a focus at (0,-4) and minor axis 3 units long?

Question

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Answer 1

General form of an ellipse is (x-x_o)^2/a^2 + (y-y_o)^2/b^2 = 1

So ( x_o, y_o) is the center, which is (0,0)

So one focus is at (0,-4) which implies this axis is the major and of value 4 and along the x-axis. The minor is of value 3 as given. It is also along the y-axis.

Need to solve for a using the two eccentricity equations.

e = c/a where c is the distance to the focus from the origin.

e = sqrt(1-b^2/a^2)

e = 4/a = sqrt(1-9/a^2) squaring both sides

16/a^2 = 1-9/a^2 and solve for a

25/a^2 = 1 so a^2 = 25 so a = 5 (discard the negative value)

So the equation is:

x^2/25 + y^2/9 = 1

Answer 2

The distance from the center of the ellipse to a focus is c:

c = | - 4 = 0 | = 4
c² = 16

The line thru the center and focus is vertical, so the major axis is also vertical.

The minor axis is 3.

2b = 3
b = 3/2
b² = 9/4

The semi-major axis can be found:

a² = b² + c²
a² = 9/4 + 16 = (9 + 64)/4 = 73/4

The equation of the ellipse is:

x²/b² + y²/a² = 1

x²/(9/4) + y²/(73/4) = 1