Find the equation of an ellipse centered at the origin with foci at the points( 0 , 3 ) and ( 0 , − 3 ). Additionally, the major axis of this ellipse has a length of 10. 15 POINTS, DUE IN 2 HOURS, will give BRAINLIEST!

Step-by-step explanation:

We know that the foci lies on the major axis, so it should lie on the x - axis. Additionally the center is the midpoint of the line joining the foci, at (0,0).

Now remember we have our foci at the point (0, ± 3). Our major axis length is 10, so if covers 5 units on either side of the x - axis. Therefore, c = 3, and a = 5. But remember that c is not part of an ellipse equation. Take a look at the formula below,

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

However we can solve for " b " instead, using c. Take a look at another major formula below,

c^2 = a^2 - b^2,

(3)^2 = (5)^2 - b^2,

9 = 25 - b^2,

- b^2 = 16,

b = 4 = - 4

Whether b = - 4 or b = 4, or equation will be the same.

(x - 0)^2 / (5)^2 + (y - 0)^2 / (4 or - 4)^2 = 1,

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

Our solution is hence option number 1 :

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

the first option is the answer.