Example 1: An arch is in the form of a semi-ellipse, and it is feet wide at the base and has a height of feet. How wide is the arch at the height of feet above the base?
Here the length of the major axis is given as and the height of the semi-ellipse is given as .
Now the standard equation of an ellipse is
Putting the values of and in the above equation of an ellipse, we have
For the height of 10 feet, the equation becomes
Thus, the required arch length is .
Example 2: An asteroid has elliptical orbit with the sun at one focus. Its distance from the sun ranges from million to million miles. Write an equation of the orbit of the asteroid.
Here we have
We also have
Now using the relation to find the value of , we have
The required equation of the ellipse of the asteroid will be