Consider the ellipse ( x^ 2 a^ 2 + y^ 2 b^ 2 =1 ) with (a>b>0 ).

MATHEMATICS · Multi-part · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Consider the ellipse

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

with

a>b>0

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

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

. (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}

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