Find an equation for the ellipse that satisfies the given conditions.
Length of major axis 6, length of minor axis 4, foci on y-axis
that would be
(x/2)^2 + (y/3)^2 = 1
or
x^2/4 + y^2/9 = 1
To find the equation of an ellipse, we need to identify the necessary information: the lengths of the major and minor axes, as well as the location of the foci.
Given that the foci of the ellipse lie on the y-axis, the coordinates of the foci would be (0, c) and (0, -c), where c is the distance from the center to the foci.
Next, we use the formula to find the distance from the center to the foci, which is given by c = √(a^2 - b^2), where a is the length of the semi-major axis and b is the length of the semi-minor axis.
In this case, the length of the major axis is 6 (which means a = 3), and the length of the minor axis is 4 (which means b = 2).
Substituting the values into the formula, we get:
c = √(3^2 - 2^2)
c = √(9 - 4)
c = √5
So the distance from the center to the foci is √5.
Now that we have all the necessary information, we can write down the equation of the ellipse. The general equation for an ellipse with the center at the origin is:
x^2/a^2 + y^2/b^2 = 1
Substituting the values of a and b we previously found, we have:
x^2/3^2 + y^2/2^2 = 1
Simplifying, we get:
x^2/9 + y^2/4 = 1
Therefore, the equation of the ellipse that satisfies the given conditions is x^2/9 + y^2/4 = 1.