SOLUTION: Write an equation for each ellipse. Center at (-3,6); major axis vertical, with length 10; C=2

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Write an equation for each ellipse. Center at (-3,6); major axis vertical, with length 10; C=2       Log On


   

Question 721364: Write an equation for each ellipse.
Center at (-3,6); major axis vertical, with length 10; C=2
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
The standard forms for the equation of a ellipse are:
%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 for horizontal ellipses
and
%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1 for vertical ellipses
In these standard forms...
  • The "h" and "k" are the x and y coordinates of the center of the hyperbola.
  • The "a" is the distance from the center to a vertex on the major axis.
  • The "b" is the distance from the center to a vertex on the minor axis.

Since we've been told that the major axis is vertical, our ellipse is vertical. So from this point on, we will be using:
%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1
  • We know that the center is (-3, 6). So the h and k of the equation will be -3 and 6.
  • Since the length of the major axis is 10, the distance from the center to each vertex will be half of that, or 5. So a = 5.
  • To find "b" we will use the equation that connects the values of a, b and c in an ellipse:
    a%5E2=b%5E2%2Bc%5E2
    %285%29%5E2=b%5E2%2B%282%29%5E2
    25=b%5E2%2B4
    21=b%5E2
    sqrt%2821%29=b (We ignore the negative square root since b is a distance.)
  • Now that we have the h, k, a and b values we can write the equation:
    %28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1

    %28x-%28-3%29%29%5E2%2F%28sqrt%2821%29%29%5E2%2B%28y-%286%29%29%5E2%2F%285%29%5E2=1
    which simplifies to:
    %28x%2B3%29%5E2%2F21%2B%28y-6%29%5E2%2F25=1