Question

A man running a race course notes that the sum of the distance from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

Answer 1

Hint: Assume that S and S’ are the flag posts and point P is the man who is running a racecourse. It is given that the sum of the distance from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. We know the property that the summation of the distance from the foci and a point on the ellipse is always a constant. Using this property, we can say that the points S and S’ are the foci of the ellipse. We know that the property that the summation of the distances from the foci and a point on the ellipse is always equal to 2a. Now, compare ‘a’ with 10 and get the value of a. We also know that in an ellipse the distance between the foci is 2ae. Here, S and S’ are the foci and the distance between them is 8. Now, compare ‘2ae’ and 8 and get the value of ‘ae’. Now, using the value of ‘a’, get the value of ‘e’. Then, put the value of ‘a’ and ‘e’ in the formula, \[{{e}^{2}}=\left( 1-\dfrac{{{b}^{2}}}{{{a}^{2}}} \right)\] and get the value of ‘b’. Now, put the value of ‘a’ and ‘b’ in the standard equation of an ellipse, \[\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\] and solve it further to get the equation of the posts traced by the man.

Complete step-by-step answer:
According to the question, we have been given that a man running a race course notes that the sum of the distance from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m.
Let us assume that S and S’ are the flag posts and point P is the man who is running a racecourse.
We have been given that the sum of the distance from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m.
Now, from the figure, we can say that
\[PS+PS'=10\] …………………………………….(1)
\[SS'=8\] …………………………….(2)
We know the property that the summation of the distance from the foci and a point on the ellipse is always a constant.
Since the sum of the distance from the two flag posts and the man is always 10 m so, we can say that S and S’ are the foci of the ellipse and point P is on the ellipse.

We also know the property that the summation of the distances from the foci and a point on the ellipse is always equal to 2a, where a is the distance on the major axis.
Using this property, we can say that \[PS+PS'=2a\] …………………………………(3)
But, from equation (1), we also have the value of \[\left( PS+PS' \right)\] .
Now, on comparing equation (1) and equation (3), we get
\[\Rightarrow 10=2a\]
\[\Rightarrow 5=a\] ……………………………………(4)
We also know that in an ellipse the distance between the foci is 2ae.
From equation (2), we have the distance between the foci of the ellipse. So,
\[\Rightarrow 2ae=8\]
\[\Rightarrow ae=4\] ……………………………(5)
Now, from equation (4) and equation (5), we get
\[\Rightarrow 5e=4\]
\[\Rightarrow e=\dfrac{4}{5}\] …………………………………(6)
We know the formula, \[{{e}^{2}}=\left( 1-\dfrac{{{b}^{2}}}{{{a}^{2}}} \right)\] …………………………………..(7)
Now, from equation (4), equation (6), and equation (7), we get
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{4}{5} \right)}^{2}}=1-\dfrac{{{b}^{2}}}{{{5}^{2}}} \\
 & \Rightarrow \dfrac{{{b}^{2}}}{25}=1-\dfrac{16}{25} \\
 & \Rightarrow \dfrac{{{b}^{2}}}{25}=\dfrac{25-16}{25} \\
 & \Rightarrow {{b}^{2}}=9 \\
\end{align}\]
\[\Rightarrow b=3\] ……………………………….(9)
We know the standard equation of the ellipse, \[\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\] …………………………………….(10)
From equation (4) and equation (9), we have the values of a and b.
Now, putting the values of a and b in equation (10), we get
\[\Rightarrow \dfrac{{{x}^{2}}}{{{5}^{2}}}+\dfrac{{{y}^{2}}}{{{3}^{2}}}=1\]
\[\Rightarrow \dfrac{{{x}^{2}}}{25}+\dfrac{{{y}^{2}}}{9}=1\] ………………………………(11)
Therefore, the equation of the posts traced by the man is \[\left( \dfrac{{{x}^{2}}}{25}+\dfrac{{{y}^{2}}}{9}=1 \right)\] .

Note: In this question, one might get confused because we don’t have any clue which directly hints towards the posts traced by the man. But we have one information that the sum of the distance from the two flag posts from him is always 10 m. It means that the flag posts are the foci of the ellipse. Here, we can say that the hidden information is that the man is running on the elliptical path.