A Note The constant which is the sum of the distances of a point on the ellipse from the two fixed points is always greater than the distance between the two fixed points. The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis . The end points of the major axis are called the vertices of the ellipse(Fig .
). MATHEMATICS . . Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse (Fig .
). Take a point P at one end of the major axis. Sum of the distances of the point P to the foci is F P + F P = F O + OP + F P (Since, F P = F O + OP) = c + a + a – c = a Take a point Q at one end of the minor axis. Sum of the distances from the point Q to the foci is F Q + F Q = c b c b c b Since both P and Q lies on the ellipse.
By the definition of ellipse, we have c b = a , i.e., a = c b or a = b + c , i.e., c = b . . Special cases of an ellipse In the equation c = a – b obtained above, if we keep a fixed and vary c from to a , the resulting ellipses will vary in shape. Case (i) When c = , both foci merge together with the centre of the ellipse and a = b , i.e., a = b , and so the ellipse becomes circle (Fig11.
). Thus, circle is a special case of an ellipse which is dealt in Section . . Case