Mathematics

Problem

Consider the ellipse

\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1

with

a>b>0

. a. (i) Define the latus rectum of an ellipse. b. (ii) Prove that the length of the latus rectum of the above ellipse is

\dfrac{2b^{2}}{a}

. c. (iii) Show that the coordinates of the end-points of the latus rectum corresponding to the focus

(c,0)

are

\bigl(c,\pm\dfrac{b^{2}}{a}\bigr)

, where

c^{2}=a^{2}-b^{2}

. d. (iv) Verify that each of the points obtained in part (iii) satisfies the equation of the ellipse.