The first step in finding the equation of an ellipse given the vertices and a focus is to determine the lengths of the major and minor axes.
The vertices (-3,-7) and (-3,3) are the endpoints of the major axis. The distance between these points will calculate the length of the major axis, which can be found using the formula |Y2 - Y1| = |3 - (-7)| = 10 units. This value is 2a, where 'a' represents half the length of the major axis, so a = 10/2 = 5.
Next, we need to find the center of the ellipse. Since the major axis goes through the center, the midpoint of the vertices will give us the coordinates of the center. The midpoint formula can be used here: (-3, (3-7)/2) = (-3, -2).
The value of 'c', which is the distance from the center to a focus, is the absolute value of the difference in the y-coordinates of the center and the focus, which yields |-2 - 2| = 4 units.
The Pythagorean Relation for any ellipse is given by a^2 = b^2 + c^2, where 'b' is half of the length of the minor axis. We already know the values of 'a' and 'c', so we can use this formula to find the value of 'b' by rearranging it as such: b = sqrt(a^2 - c^2) = sqrt(25 - 16) = 3.
Now that we know the values of 'a', 'b', and the coordinates of the center, we can define the equation of the ellipse. In the general form of the equation of an ellipse with its center at (h, k), the equation is: ((x - h)^2/a^2) + ((y - k)^2/b^2) = 1.
Therefore, the equation of the ellipse with its center at (-3, -2), a = 5, and b = 3 is ((x+3)^2)/5^2 + ((y+2)^2)/3^2 = 1 or, simplifying the squares, ((x+3)^2)/25 + ((y+2)^2)/9 = 1. This is the equation of the ellipse.
For graphing this ellipse, plot the center at (-3, -2), vertices at (-3,-7) and (-3,3), then draw the ellipse using the lengths of the major and minor axes. The focus point (-3,2) should lie inside the ellipse along the major axis.