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Some Important Facts and Definitions about Ellipse

  1. As 0 < e < 1, V2 will be on M1 F1 [and not F1M1 produced].
  2. Symmetry
    The equation of the ellipse contains only even powers of x and y. Therefore, if (x, y) is any point on the curve, then (- x, y) is also a point on it. In other words the curve is symmetrical about y-axis. Similarly, the curve is symmetrical about x-axis, i.e. if (x, y) is any point on the curve, then (x, -y) is also a point on it.
  3. Intersection with the axis
    The ellipse meets x-axis when y=0, i.e. when . Therefore, the ellipse meets x-axis in the points V1(-a, 0) and V2(a, 0).Similarly, it meets y-axis in the points (0, b) and (0, -b).
  4. Range

    As y2 0, we get a2 - x2 0 -a x a. Therefore, no point of the curve lies to the right x = a and to the left of x = -a. Similarly, we can show that no point of the curve lies above y = b and below y =-b.
  5. The ellipse has a second focus and a second directrix
    As the ellipse is symmetrical about y-axis, therefore, if we take the points F2 and M2 on the x-axis, such that OF2 = OF1 and OM2 = OM1then the point F2 will also be a focus of the ellipse and the line through M2 and parallel y-axis will be corresponding directrix. This focus is F2 (ae, 0) and the directrix is .
  6. Principal Axes
    The lines V1V2 and B1B2 about which the curve is symmetrical are called axes of the ellipse and together they are called Principal axes.
  7. As b2 = a2 (1 - e2) and 0 < e < 1, b < a. As such V1V2 is called the major axis and B1B2, the minor axis of the ellipse.
  8. The points V1 and V2, the extremities of major axis are called the vertices of the ellipse.
  9. If (x, y) is a point on the curve, so is the point (-x, -y) and the join of these points is bisected at O (0, 0). Thus, every chord of the ellipse which passes through O is bisected by it. As such O is called the centre of the ellipse. A chord through O, the centre, is called a diameter of the ellipse.
  10. If P is a point on the ellipse, then F1P and F2P are called focal distances of P and a chord through any one of the foci is called a focal chord.
  11. The eccentricity measures the flatness of the ellipse. The distance between two foci F1 and F2 is 2ae. As e increases, the distance between foci increases, and the foci F1 and F2 move away from the centre and the ellipse become flatter. To see it in another way we note that b2 = a2 - a2 e2 = a2 (1 - e2) becomes smaller if a is kept constant and e becomes larger. Hence, the ellipse becomes flatter if e becomes larger. On the other hand, if e 0 (i.e. e approaches zero), the foci come closer to the centre and a2 - b2 = a2 e2 0, b 0. In other words the ellipse looks almost like a circle. In fact, we can treat a circle to be an ellipse with eccentricity equal to zero.
  12. If take the major axis along the y-axis, focus as (0, ae) and directrix as , the equation of the ellipse is given by
    where b2 = a2 (1 - e2) (See in figure given below.)
In this case the other focus is (0, -ae) and the other directrix is .

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