Write an equation of the ellipse with foci at (0 ±11) and vertices at (0 ±12)

When you plot the given points on a coordinate plane, you can see this is an ellipse that is vertical.  The equation for this is .  Of the values a and b, a is always greater than b, so it is on our major axis, the y-axis (it's under the y-k).  a is the distance from the center to the vertices.  When you plotted those given points, you can see that point dead center between the foci and the vertices is the origin (0, 0).  That gives us our h and k values of 0 and 0.  a is the number of units from the h coordinate of 0 to the y coordinate of the vertex, so a = 12.  c is the number of units from the center to the foci, so c = 11.  We need to find b now using the formula for the foci which is .  We have c and a, so and .  Now we have all the info we need to rewrite the equation: or, simplified,

x^2 + 81

we can factor this using difference of square binomials.

rewrite the equation:

x^2 - (-81)

factor:

(a^2 - b^2) = (√a^2 + √b^2)(√a^2 - √b^2)

(x^2 - (-81) = (x + √-81)(x - √-81)

we can remove the - from under the radical by multiplying it by i.

(x^2 - (-81) = (x + i√81)(x - i√81)

(x^2 - (-81) = (x + 9i)(x - 9i)

(x + 9i)(x - 9i)

we can check our answer by using foil:

x^2 - 9ix + 9ix - 81i^2

x^2 - 81^2

i^2 = -1

x^2 + 81