Shortest Distance between a Point and a Circle
What is the distance between a circle with equation which is centered at the origin and a point ?
The ray , starting at the origin and passing through the point , intersects the circle at the point closest to . So, the distance between the circle and the point will be the difference of the distance of the point from the origin and the radius of the circle.
Using the Distance Formula , the shortest distance between the point and the circle is .
Note that the formula works whether is inside or outside the circle.
If the circle is not centered at the origin but has a center say and a radius , the shortest distance between the point and the circle is .
Example 1:
What is the shortest distance between the circle and the point ?
The circle is centered at the origin and has a radius .
So, the shortest distance between the point and the circle is given by
That is, the shortest distance between them is units.
Example 2:
What is the shortest distance between the circle and the point ?
The circle is centered at the origin and has a radius .
So, the shortest distance between the point and the circle is given by
That is, the shortest distance between them is about units.
Example 3:
What is the shortest distance between the circle and the point ?
Compare the given equation with the standard form of equation of the circle,
where is the center and is the radius.
The given circle has its center at and has a radius of units.
Then, the shortest distance between the point and the circle is given by
That is, the shortest distance between them is about units.
Example 4:
What is the shortest distance between the circle and the point ?
Rewrite the equation of the circle in the form where is the center and is the radius.
So, the circle has its center at and has a radius of units.
Then, the shortest distance between the point and the circle is given by
That is, the shortest distance between them is units.