The equation of a vertical ellipse is x^2b^2+ y^2a^2=1. The equation of a horizontal ellipse is x^2a^2+ y^2b^2=1. In both cases a and b are positive numbers, with a>b.
x^2/72.25+y^2/90.25=1
Practice makes perfect
Since the given focus is on the y-axis, the ellipse is vertical. Recall the general equation for this type of ellipse.
x^2/b^2+y^2/a^2=1, a>b>0
Here, the vertices are (0, ± a) and the co-vertices (± b,0). Let's think about how a relates to the height of a vertical ellipse centered at the origin. Because the vertices extend from the negative to the positive values of a, the length of the major axis will always be 2a. Let's look at an example!
This means we can find the value of a using the given height.
If we let (0,± c) be the foci of our vertical ellipse, we can write an equation connecting a, b, and c.
c^2=a^2-b^2
We are told that the point ( 0,3sqrt(2) ) is a focus, and therefore c=3sqrt(2). Moreover, we already found that a= 9.5. Let's substitute these values into the above equation to find the value of b.
Note that we only took the principal root because b is the absolute value of the nonzero coordinate of the co-vertices. Now that we know that b= 8.5, we can finally write the equation of the ellipse.
x^2/8.5^2+y^2/9.5^2=1 ⇔ x^2/72.25+y^2/90.25=1
