The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Shortest Distance between a Point and a Circle

Date: 1/22/96 at 10:14:18
From: Anonymous
Subject: Calculus

Problem:  Find the shortest distance between the point (4,5) and
the circumference of x^2 + y^2 = 9.

The problem involves finding the derivative and then letting that
equal zero to find the minimum.  Where I'm stuck is I don't know 
how to set up the relationship between the circle and the point so 
that I get an equation to derive.

Please help!

Date: 5/30/96 at 14:54:23
From: Doctor Charles
Subject: Re: Calculus

Well you CAN use calculus.  Here's the technique:

Use polar coordinates to cut down x and y to one variable.  
Any point on the circle has coordinates (3 cos theta, 3 sin theta) for 
some theta. 

Then let L be the square of the distance of (4,5) from this point:

   L = (3*(cos theta) - 4)^2 + (3*(sin theta) - 5) ^2

Using (cos^2 theta + sin^2 theta = 1) you should get:

   L = 50 - 24 cos theta - 30 sin theta

Differentiate with respect to theta and get that tan theta = 4/5.  This 
will actually give you two values of theta.  One is the point on the 
circle furthest from (4,5) and the other gives you the one nearest (4,5). 
Substitute these into the formula of L to get the square of the 
distances.  Choose the smallest!

Without using calculus:

Note that (4,5) is sqrt(4^2 + 5^2) units away from the origin. The 
circle is the set of points 3 units from the origin.  If you draw a line 
from the origin to (4,5) then it will cut the circle at the point 
closest to (4,5). The length of the line from the circle to (4,5) must 

        sqrt(4^2 + 5^2) - 3 = sqrt(41) - 3

The corresponding point on the circle has coordinates (a,b) given by:

        (a,b) = (3/sqrt(41)*4,3/sqrt(41)*5)
              = (12/sqrt(41) , 15/sqrt(41))

This is much simpler than the calculus method, although both will give 
the same answer.

-Doctor Charles,  The Math Forum

Associated Topics:
High School Calculus

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.