A Note The standard equations of ellipses have centre at the origin and the major and minor axis are coordinate axes. However, the study of the ellipses with centre at any other point, and any line through the centre as major and the minor axes passing through the centre and perpendicular to major axis are beyond the scope here. From the standard equations of the ellipses (Fig11. ), we have the following observations: .
Ellipse is symmetric with respect to both the coordinate axes since if ( x , y ) is a point on the ellipse, then (– x , y ), ( x , – y ) and (– x , – y ) are also points on the ellipse. . The foci always lie on the major axis. The major axis can be determined by finding the intercepts on the axes of symmetry.
That is, major axis is along the x -axis if the coefficient of x has the larger denominator and it is along the y -axis if the coefficient of y has the larger denominator. MATHEMATICS . . Latus rectum Definition Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse (Fig .
). To find the length of the latus rectum of the ellipse b Let the length of AF be l . Then the coordinates of A are ( c , l ) , i.e., ( ae , l ) Since A lies on the ellipse b = , we have ae l b ⇒ l = b ( – e ) But c a – b b e – Therefore l = b , i.e., b l Since the ellipse is symmetric with respect to y -axis (of course, it is symmetric w.r.t. both the coordinate axes), AF = F B and so length of the latus rectum is b a .
Example Find the coordinates of the foci, the vertices, the length of major