In ellipse geometry, the first eccentricity e is defined as e = sqrt(1 - (b^2 / a^2)). Which option correctly shows this relation?

• A. e = sqrt(1 - (b^2 / a^2))
• B. e = a/b
• C. e = b/a
• D. e = a + b

Answer: (A)

In ellipse geometry, eccentricity measures how stretched the ellipse is and is defined as e = c/a, where c is the distance from the center to a focus and a is the semi-major axis. Since c^2 = a^2 − b^2, substituting gives e = sqrt(1 − b^2/a^2). This form directly shows why e lies between 0 and 1 for ellipses, with e = 0 for a circle (a = b) and e approaching 1 as the ellipse becomes very elongated. The expression matches this standard relationship exactly, so it is the correct representation. The other forms do not reflect this derivation and do not produce a dimensionless value between 0 and 1 in general.