k , and the coordinates of the foci are h c k and h c k where c (b) Major axis parallel to the y -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 k ) and h k ) , and the coordinates of the foci are h k ) and h k ) , where c - - Equation Centre Major Axis Vertices Foci h ) = a > Fig. . (a) Major axis parallel to the x -axis Foci are c units right and c units left of centre, where c h k parallel to the x -axis h a k h a k h c k h c k h > x' y' A(h,k+a) S(h,k+c) O C ( h , k ) S' ( h,k–c ) A' ( h,k–a ) Fig. .
(b) Major axis parallel to the y -axis Foci are c units right and c units left of centre, where c h k parallel to the y -axis ( , h k ( , h k ( , h k ( , h k Theorem . The sum of the focal distances of any point on the ellipse is equal to length of the major axis. Proof Let P x y ( , ) be a point on the ellipse x = . Draw MM ′ through P , perpendicular to directrices l and ′ l .
Draw PN ⊥ to x -axis. By definition SP = ePM = eNZ = e CZ CN ] Fig. . S' (– c , ) S ( c , ) P ( x,y ) d d M Z(a/e , ) Z'(–a/e , ) M' l l' N C ( , ) C ( h , k ) O S(h+c,k) S'(h–c,k) A(h+a,k) A ' (h –a,k) y' x' - - Two Dimensional Analytical Geometry - II = e a e ex =