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 we have length of major axis is given as and height of a semi-ellipse is given as .
Now standard equation of ellipse is
Putting the values of and in the above equation of ellipse, we have
For the height of 10 feet, the equation becomes
Thus, the required arch length is .
Example 2: An asteroid has elliptic 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
Also we have
Now using the relation to find the value of , we have
Required equation of ellipse of asteroid will be