Mathematics
Write the equation of 2 different ellipses...

### Question Description

which each have foci at (-3, 0) and (3, 0).

The relationship between an ellipse and its foci is that the total distance from any point on an ellipse to the two foci is a constant. There are infinite possible ellipses with the foci (-3,0) and (3,0), each with a different distance (i.e. bigger and smaller ellipses!). So to find two let's just pick some values for that distance!

Remember that for an ellipse, the distance from each focus to the centre (which is (0,0) here) is given by the formula:

F^2 =  j^2 - n^2

where F is the distance from the focus to the centre, a is the major axis radius, and b is the minor axis radius. We know that our foci are 3 units from the centre at (0,0) so we have to find a and b such that a >= b and:

3^2 = a^2 - b^2

a^2 - b^2 = 9

So if we pick a = 5, then

b^2 = 5^2 - 9

= 25-9

= 16, and b = 4.

We know our ellipse is horizontal (the foci lie on the x-axis) so using the equation for an ellipse:

x^2 / a^2 + y^2 / b^2 = 1

We have:

x^2 / 5^2 + y^2 / 4^2 = 1

x^2 / 25 + y^2 / 16 = 1

The rest of our options won't have nice integers, but since we just need to put a^2 and b^2 into the equation, and we know a^2 - b^2 = 9, we can pick any two numbers for which that's true. For example, a^2 = 20 and b^2 = 11 (so a slightly smaller ellipse):

x^2 / 20 + y^2 / 11 = 1

Hope this was helpful for you!

iainharlow (1041)
Cornell University
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