# Applications of Ellipse

__Example 1__**:** An arch is in the form of a semi-ellipse, and it is $$48$$ feet wide at the base and has a height of $$20$$ feet. How wide is the arch at the height of $$10$$ feet above the base?

Here the length of the major axis is given as $$2a = 48\,\, \Rightarrow a = 24$$ and the height of the semi-ellipse is given as $$b = 20$$.

Now the standard equation of an ellipse is

\[\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\]

Putting the values of $$a$$ and $$b$$ in the above equation of an ellipse, we have

\[\frac{{{x^2}}}{{{{\left( {24} \right)}^2}}} + \frac{{{y^2}}}{{{{\left( {20} \right)}^2}}} = 1\]

For the height of 10 feet, the equation becomes

\[\begin{gathered} \frac{{{x^2}}}{{{{\left( {24} \right)}^2}}} + \frac{{{{\left( {10} \right)}^2}}}{{{{\left( {20} \right)}^2}}} = 1 \\ \Rightarrow \frac{{{x^2}}}{{576}} + \frac{{100}}{{400}} = 1\,\,\, \Rightarrow \frac{{{x^2}}}{{576}} + \frac{1}{4} = 1 \\ \Rightarrow \frac{{{x^2}}}{{576}} = \frac{3}{4} \\ \Rightarrow x = \pm 12\sqrt 3 \\ \end{gathered} \]

Thus, the required arch length is $$24\sqrt 3 $$.

__Example 2__**:** An asteroid has elliptical orbit with the sun at one focus. Its distance from the sun ranges from $$18$$ million to $$182$$ million miles. Write an equation of the orbit of the asteroid.

Here we have

\[\begin{gathered} {V_2}{F_2} = 18 = {V_1}{F_1} \\ {V_1}{F_2} = 182 = {V_2}{F_1} \\ \end{gathered} \]

We also have

\[\begin{gathered} 2ae = |{F_1}{F_2}| = |{V_1}{F_2} – {V_1}{F_1}| = 182 – 18 = 164 \\ \Rightarrow ae = 82 \\ a = |C{V_2}| = |C{F_2}| – |{F_2}{V_2}| = 82 + 18 = 100 \\ \end{gathered} \]

Now using the relation to find the value of $$b$$, we have

\[\begin{gathered} {\left( {ae} \right)^2} = {a^2} – {b^2} \\ \Rightarrow {b^2} = {\left( {100} \right)^2} – {\left( {82} \right)^2} = 3276 \\ \end{gathered} \]

The required equation of the ellipse of the asteroid will be

\[\begin{gathered} \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1 \\ \Rightarrow \frac{{{x^2}}}{{10000}} + \frac{{{y^2}}}{{3276}} = 1 \\ \end{gathered} \]