To find the equation of an ellipse, we first start by understanding that the vertices we have are at (5, 0) and (-5, 0), meaning this ellipse is centered around the origin, (0, 0), and its major axis is along the x-axis.
The distance from the center of the ellipse to either of its vertices is defined as 'a', which becomes the radius along the major axis. Given the coordinates of the vertices, we can see that the x-coordinates are 5 and -5. Therefore, the length of the major axis is 2a, which equals to 10. That means a is 5.
Secondly, the length of the minor axis is given as 6. Since the minor axis extends from the center of the ellipse to either of the minor vertices, this distance is defined as 'b'. Given that the length of the minor axis is 2b, we can deduce that b is 6/2, which equals 3.
Now, let's recall the standard formula for an ellipse that is centered at the origin:
(x^2 / a^2) + (y^2 / b^2) = 1
Substituting the values we've found for 'a' and 'b' into this equation, we get our final equation
(x^2 / 5^2) + (y^2 / 3^2) = 1,
which simplifies to:
(x^2 / 25) + (y^2 / 9) = 1.
The equation for the ellipse with the given vertices and minor axis length is (x^2 / 25) + (y^2 / 9) = 1.