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

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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 - b. Substituting c2 = 12, or 1,  and a2 = 52, or 25, we get: 1 = 25-b2, or b2 = 25-1, or b2 = 24.
 With a= 25, b2 = 24 the equation of this ellipse can be written as
                         x2/25 + y2/24 = 1