Question

In an ellipse, the distance between its foci is 6 and the minor axis is 8. Then its eccentricity is:
A. $ \dfrac{3}{5} $
B. $ \dfrac{1}{2} $
C. $ \dfrac{4}{5} $
D. $ \dfrac{1}{{\sqrt 5 }} $

Answer 1

Hint: We know that the eccentricity of an ellipse can be defined as the ratio of its linear eccentricity to the length of the semi major axis. Here, linear eccentricity is the distance of the focal point to the center of the ellipse. We will use this definition along with the equation of ellipse to find the eccentricity of the given ellipse.

Complete step-by-step answer:

We know that the equation of ellipse is given by
 $ \dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1,a > b $
Here, we are given that the minor axis is 8.
 $ \Rightarrow 2b = 8 $
As per the definition of eccentricity of the ellipse,
 $ e = \dfrac{c}{a} $ , where, $ c $ is the linear eccentricity and $ a $ is the semi major axis,
Also as per the definition of linear eccentricity,
 $ c = \dfrac{f}{2} $ , Where, $ f $ is the distance between foci of an ellipse
Putting the value of $ c $ in the equation of eccentricity, we get
\[
  e = \dfrac{f}{{2a}} \\
   \Rightarrow 2ae = f \]
But, we are given that the distance between foci of the ellipse is 6
\[ \Rightarrow 2ae = 6\]
Now, by dividing \[2b = 8\] by \[2ae = 6\], we get
\[
  \dfrac{{2b}}{{2ae}} = \dfrac{8}{6} \\
   \Rightarrow \dfrac{b}{{ae}} = \dfrac{4}{3} \]
Squaring both the sides,
\[
   \Rightarrow \dfrac{{{b^2}}}{{{a^2}{e^2}}} = \dfrac{{16}}{9} \\
   \Rightarrow \dfrac{{{b^2}}}{{{a^2}}} = \dfrac{{16}}{9}{e^2} \]
 We know that for an ellipse \[\dfrac{{{b^2}}}{{{a^2}}} = 1 - {e^2}\]
\[
   \Rightarrow 1 - {e^2} = \dfrac{{16}}{9}{e^2} \\
   \Rightarrow 1 = \dfrac{{16}}{9}{e^2} + {e^2} \\
   \Rightarrow 1 = \dfrac{{16 + 9}}{9}{e^2} \]
\[
   \Rightarrow 1 = \dfrac{{25}}{9}{e^2} \\
   \Rightarrow {e^2} = \dfrac{9}{{25}} \]
\[ \Rightarrow e = \dfrac{3}{5}\]
Thus, the eccentricity of the given ellipse is $ \dfrac{3}{5} $ .
So, the correct answer is “Option A”.

Note: Here, we have used the formula for the eccentricity of an ellipse according to its definition, $ e = \dfrac{c}{a} $ . It is important to know that eccentricity of the ellipse is always between 0 and 1. Also, eccentricity of the circle is 0, eccentricity of the parabola is 1 and eccentricity of hyperbola is greater than 1.