Write an equation for the ellipse given only the center (-1,1)

Question

Book answer is (x-1)²/4 + (y-1)²=1, but I have no idea where the 4 came from because I am not given foci or vertices. The major axis is on the x-axis (horizontal on the graph)

Answer 1

For an ellipse centered at (1,-1)

with a major axis of length 2a lying along the horizontal X-axis

and a minor axis of length 2b lying along the vertical Y--axis,

the equation for this ellipse is:

(x-1)² / a² + (y+1) / b² = 1

Combining the fractions, this can be rewritten as:

b² (x-1)² + a² (y+1) = a² b²

However, if the equation of an ellipse is given as:

(x-1)² / 4 + (y-1)² = 1

then the center is at (1,1) -- not (-1,1) .

The semimajor & semiminor lengths are 2 & 1, respectively,

so the ends of the major axis (vertices) are at (3,1) and (-1,1)

while the ends of the minor axis are at (1,2) and (1.0)

Therefore, the "Book answer" (x-1)²/4 + (y-1)²=1

is not consistent with the statement that the center is at (-1,1)