# Some Important Facts and Definitions about Ellipse

- As 0 < e < 1, V
_{2}will be on M_{1 }F_{1}[and not F_{1}M_{1}produced]. **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.**Intersection with the axis**

The ellipse meets x-axis when y=0, i.e. when . Therefore, the ellipse meets x-axis in the points V_{1}(-a, 0) and V_{2}(a, 0).Similarly, it meets y-axis in the points (0, b) and (0, -b).**Range**

As y^{2}â‰¥ 0, we get a^{2 }- x^{2}â‰¥ 0 â‡’ -a â‰¤ x â‰¤ 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.**The ellipse has a second focus and a second directrix**

As the ellipse is symmetrical about y-axis, therefore, if we take the points F_{2}and M_{2}on the x-axis, such that OF_{2 }= OF_{1}and OM_{2 }= OM_{1}then the point F_{2}will also be a focus of the ellipse and the line through M_{2}and parallel y-axis will be corresponding directrix. This focus is F_{2 }(ae, 0) and the directrix is .**Principal Axes**

The lines V_{1}V_{2}and B_{1}B_{2}about which the curve is symmetrical are called axes of the ellipse and together they are called Principal axes.- As b
^{2 }= a^{2 }(1 - e^{2}) and 0 < e < 1, b < a. As such V_{1}V_{2}is called the major axis and B_{1}B_{2}, the minor axis of the ellipse. - The points V
_{1}and V_{2}, the extremities of major axis are called the vertices of the ellipse. - 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.
- If P is a point on the ellipse, then F
_{1}P and F_{2}P are called focal distances of P and a chord through any one of the foci is called a focal chord. - The eccentricity measures the flatness of the ellipse. The distance between two foci F
_{1}and F_{2}is 2ae. As e increases, the distance between foci increases, and the foci F_{1}and F_{2}move away from the centre and the ellipse become flatter. To see it in another way we note that b^{2 }= a^{2 }- a^{2}e^{2 }= a^{2 }(1 - e^{2}) 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 â†’ 0 (i.e. e approaches zero), the foci come closer to the centre and a^{2 }- b^{2 }= a^{2 }e^{2 }^{â†’ }0, b â†’ 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. - 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 b^{2 }= a^{2 }(1 - e^{2}) (See in figure given below.)

â€‹â€‹â€‹

In this case the other focus is (0, -ae) and the other directrix is .