Write an equation for the ellipse with end points of the major axis at (7,1) and (-7,1) and end points of the minor axis at (0,5) and (0,-3).

I do not know how to start this. There are no examples in my book that I can study.

Could you show me how to set this up?

Thanks.

The major axis is along y = 1 and the minor axis is along x = 0.

The center is at x=0 and y=1

The major axis is a = [7 - (-7)]/2 = 7 and the minor axis is b = [5 -(-3)]/2 = 4

Can you complete the answer?
(x/a)^2 + [(y-1)/b]^2 = 1

Sure! To find the equation of an ellipse, we can start by noting that the center of the ellipse is the midpoint between the two endpoints of the major axis, and the major and minor axes are perpendicular to each other.

Let's begin by finding the center of the ellipse. The midpoint formula can be used to find the coordinates of the center, so let's calculate that first:

Midpoint of major axis:
x-coordinate of center = (x1 + x2) / 2 = (7 + (-7)) / 2 = 0
y-coordinate of center = (y1 + y2) / 2 = (1 + 1) / 2 = 1

Therefore, the center of the ellipse is (0, 1).

Next, we need to determine the lengths of the major and minor axes. The length of the major axis is the distance between the two endpoints, and similarly, the length of the minor axis is the distance between the two endpoints:

Length of major axis = distance between (7, 1) and (-7, 1)
= sqrt((7 - (-7))^2 + (1 - 1)^2)
= sqrt((14)^2 + (0)^2)
= 14

Length of minor axis = distance between (0, 5) and (0, -3)
= sqrt((0 - 0)^2 + (5 - (-3))^2)
= sqrt((0)^2 + (8)^2)
= 8

Now, we have the center and the lengths of both the major and minor axes, which are essential to defining the ellipse's equation.

The standard equation of an ellipse with a center at (h, k), major axis of length 2a, and minor axis of length 2b is:

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

Plugging in the values we have:

(x - 0)^2/7^2 + (y - 1)^2/4^2 = 1

Simplify and rewrite the equation:

x^2/49 + (y - 1)^2/16 = 1

So, the equation of the given ellipse is x^2/49 + (y - 1)^2/16 = 1.

I hope this helps! Let me know if you have any further questions.