If the distance between the foci is equal to the length of the latus rectum, then the eccentricity of the ellipse is

Step-by-step solution

For an ellipse with semi-major axis a and semi-minor axis b , the distance between the foci is 2ae and the length of the latus rectum is 2b^2 a . Given that the distance between the foci is equal to the length of the latus rectum, we have 2ae = 2b^2 a . Simplifying this, ae = b^2 a , which implies a^2e = b^2 . Using the relation b^2 = a^2(1 - e^2) , we substitute b^2 into the equation: a^2e = a^2(1 - e^2) e = 1 - e^2 e^2 + e - 1 = 0 Solving this quadratic equation for e using the quadratic formula e = -b b^2 - 4ac