Question

Why is the eccentricity of an ellipse between $0$and $1$ ?

Answer 1

Hint: In mathematics, an ellipse is a plane curve surrounding two focal points, such that fir all points on the curve, the sum of the two distances to the focal points is a constant. The shape of an ellipse is in oval shape it has two axes namely the major axis and the minor axis. The length of the major axis is $2a$ when $x$ - axis is the major axis and the length of the minor axis is $2b$ when $y$- axis is the minor axis. Or it can even be vice-versa.

Complete step-by-step answer:
The general equation of the ellipse is $\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1$ .
If $a>b$ , then $x$ - axis is the major axis.
If $aIt looks like this :

The ratio of distance from the center of the ellipse from either focus to the semi-major axis of the ellipse is defined as the eccentricity of the ellipse.
The eccentricity of the $e=\dfrac{c}{a}$ where $c$is the focal length and $a$is the length of the semi major axis.
We know that $c\le a$ , and that is why eccentricity is always less than $1$ and greater than $0$ .
We also know that ${{c}^{2}}={{a}^{2}}-{{b}^{2}}$ , therefore eccentricity becomes the following :
$\begin{align}
  & \Rightarrow e=\dfrac{c}{a} \\
 & \Rightarrow e=\dfrac{\sqrt{{{a}^{2}}-{{b}^{2}}}}{a} \\
 & \Rightarrow e=\sqrt{\dfrac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}}} \\
\end{align}$
If $b$ is the length of the semi-major axis, then eccentricity would be the following :
$\begin{align}
  & \Rightarrow e=\dfrac{c}{a} \\
 & \Rightarrow e=\dfrac{\sqrt{{{b}^{2}}-{{a}^{2}}}}{b} \\
 & \Rightarrow e=\sqrt{\dfrac{{{b}^{2}}-{{a}^{2}}}{{{b}^{2}}}} \\
\end{align}$

Note: It is very important to remember all the formulae and definitions relating to the ellipse. It is a very important conic. All the definitions and the formulae of the other conics such as circles, parabolas and hyperbolas must also be remembered since all these concepts can be clubbed and asked as one question in the exam. It is important to note the differences in the formulae when the major and minor axes are being reversed.