find the equation of an ellipse with a center at (0,0) and focus at (1,0) and vertex at (5,0)

Because the center of the ellipse is at the origin and a focus is on the x-axis the foci can be  written as (c,0) and (-c,0). Therefore c=1 and the major axis is on the x-axis which means the standard form of this ellipse will be in this form:  (x-h)²/a² + (y-k)²/b² = 1 where h and k are the x and y co-ordinates of the center point which is (0,0). Simplifying:  x²/a²  +  y²/b² = 1. Also, "a" equals the length from the center of the ellipse to the vertex which, in this case, is 5, and c² = a² - b² . Substituting c² = 1², or 1,  and a² = 5², or 25, we get: 1 = 25-b², or b² = 25-1, or b² = 24.

 With a² = 25, b² = 24 the equation of this ellipse can be written as

 

                         x²/25 + y²/24 = 1