How do you write an equation of an ellipse in standard form given Vertices: (-9,18), (-9,2), Foci (-9,14), (-9,2)?

1 Answer
Jan 28, 2017

Answer:

The foci are #(h,k-sqrt(a^2-b^2)) and (h,k+sqrt(a^2-b^2))# for an ellipse with a vertical major axis. The vertices are #(h,k-a) and (h,k+a)#
The equation is #(y-k)^2/a^2+(x-h)^2/b^2=1" [1]"#

Explanation:

Use the given vertices to write 3 equations:

#h = -9" [2]"#
#k-a=2" [3]"#
#k+a=18" [4]"#

To find the value of k, add equations [3] and [4]:

#2k = 20#

#k = 10#

To find the value of a, substitute 10 for k into equation [4]:

#10+a=18#

#a = 8#

Use #a = 8, k = 10# and the y coordinate of one of the foci to write an equation where b is the only unknown:

#14 = 10 + sqrt(8^2 - b^2)#

#4 = sqrt(8^2 - b^2)#

#16 = 8^2 - b^2#

#-48 = -b^2#

#b = 4sqrt(3)#

Substitute, #h = -9,k=10, a = 8, and b = 4sqrt3# into equation [1]:

#(y-10)^2/8^2+(x- -9)^2/(4sqrt3)^2=1" "larr# the answer.