Question

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is \[\dfrac{2}{3}\] then the eccentricity of the ellipse is:
A. \[\dfrac{{2\sqrt 2 }}{3}\]
B. \[\sqrt 5 \]
C. \[8\]
D. \[2\]

Answer 1

Hint: In order to solve the question given above, we will be looking back at the concepts of ellipse. All the properties of ellipse will be used to solve this question. Some of them are the length of the major axis, length of the minor axis, area of the ellipse, etc. You also need to remember the area of a circle.

Formula used:
To solve this question, you need to remember certain properties of ellipse.
The length of the major axis \[ = 2a\].
The length of the minor axis \[ = 2b\].
The area of the circle \[ = \pi {r^2}\] .
The area of the ellipse \[ = \pi ab\].
\[P = \dfrac{{\pi ab - \pi {r^2}}}{{\pi ab}}\] .
\[e = \sqrt {1 - {{\left( {\dfrac{a}{b}} \right)}^2}} \] .

Complete step by step solution:
We are given that the probability that the point lies outside the ellipse is \[\dfrac{2}{3}\] .
Let the radius of the circle \[ = a\] .
We know that length of the major axis \[ = 2a\] and that of the minor axis \[ = 2b\] .
We also know that the area of the circle \[ = \pi {r^2}\] and the area of the ellipse \[ = \pi ab\] .
We will use these values to find our answer.
We know, \[P = \dfrac{{\pi ab - \pi {r^2}}}{{\pi ab}}\]
\[ = \dfrac{{\pi ab - \pi {a^2}}}{{\pi ab}}\]
This can be written as,
\[
  P = 1 - \dfrac{a}{b} \\
   \Rightarrow \dfrac{a}{b} = 1 - \dfrac{2}{3} \\
   \Rightarrow \dfrac{a}{b} = \dfrac{1}{3} \\
 \] .
Now, we will find the eccentricity.
\[e = \sqrt {1 - {{\left( {\dfrac{a}{b}} \right)}^2}} \]
    \[
   = \sqrt {1 - {{\left( {\dfrac{1}{3}} \right)}^2}} \\
   = \sqrt {\dfrac{8}{9}} \\
 \].
This can be written as \[\dfrac{{2\sqrt 2 }}{3}\] .

So, the correct answer is Option A.

Note: While solving sums similar to the one given above, you need to remember the concepts of ellipse. In the above question, we have used several properties of ellipse such as the properties to calculate the length of the major and the minor axis and the formula to calculate the area of the ellipse. We have also used a formula while finding the eccentricity.