. ( ) The line segment BB ′ is called the minor axis of the ellipse and is of length b . ( ) The line segment CA = the line segment CA ′ = semi major axis = a and the line segment CB = the line segment CB ′ = semi minor axis = b . ( ) By symmetry, taking ′ − S ae , ) as focus and x e as directrix ′ l gives the same ellipse.
Thus, we see that an ellipse has two foci, S ae , ) and ′ − S ae , ) and two vertices A a ( , ) and ′ − , ) and also two directrices, x e and x e Fig. . P ( x,y ) M Z C ( , ) S ( ae , ) ′ L a ( , l A( a, ) L ( a, b ) ′ Z ′ l S' (– ae , ) A' (– a , ) B' B Centre Latus rectum Foci Vertices - - Two Dimensional Analytical Geometry - II Example . Find the length of Latus rectum of the ellipse x = .
The Latus rectum LL ′ (Fig. . ) of an ellipse x = passes through S ae , ) . Hence L is ( ae y .
Therefore, a e = = - e y = b e ( = b since, e y = ± b That is, the end points of Latus rectum L and ′ L are ae b and ae , − Hence the length of latus rectum LL ′ = (ii) Types of ellipses with centre at ( , ) h k (a) Major axis parallel to the x -axis From Fig. . h ) + ) = > , The length of the major axis is a . The length of the minor axis is b .
The coordinates of the vertices are h a k and h a