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 we have length of major axis is given as $2a = 48\,\, \Rightarrow a = 24$ and height of a semi-ellipse is given as $b = 20$.
Now standard equation of ellipse is

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

For the height of 10 feet, the equation becomes

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

Example 2: An asteroid has elliptic 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

Also we have

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

Required equation of ellipse of asteroid will be