MATH 120 Lecture 33: Tangent lines to a circle
Tangent lines to a circle
This example will illustrate how to find the tangent lines to a given circle which pass through
a given point.
Suppose our circle has center (0,0) and radius 2, and we are interested in tangent lines to the
circle that pass through (5,3).
The picture we might draw of this situation looks like this.
(5,3)
We are interested in finding the equations of these tangent lines (i.e., the lines which pass
through exactly one point of the circle, and pass through (5,3)).
The key is to find the points of tangency, labeled A1and A2in the next figure.
(5,3)
A1
A2
The trick to doing this is to introduce variables for the coordinates for one of these points. Let’s
say one of these points is (a, b).
(Note: I strongly recommend using variables other than xand yhere, so that we can easily
remember that (a, b)is a point of tangency; xand yare too generic, and we will want to use
them later for other purposes (such as expressing our line equations).)
Then this is the key idea: Since we have two variables (i.e., unknowns), we want to come up
with two equations that these variables satisfy. If we can do that, then we can use algebra to
solve for our unknowns.
How do we come up with two equations?
First, we know that (a, b)is a point on our circle, and so (a, b)satisfies the equation of the circle.
Thus,
a2+b2= 22= 4 (1)
So we have one equation.
To get a second equation, we need to use the fact that the line through (a, b)and (5,3) is tangent
to the circle. A geometric consequence of this is that this line is perpendicular to the radius
Document Summary
This example will illustrate how to nd the tangent lines to a given circle which pass through a given point. Suppose our circle has center (0, 0) and radius 2, and we are interested in tangent lines to the circle that pass through (5, 3). The picture we might draw of this situation looks like this. (5, 3) We are interested in nding the equations of these tangent lines (i. e. , the lines which pass through exactly one point of the circle, and pass through (5, 3)). The key is to nd the points of tangency, labeled a1 and a2 in the next gure. (5, 3) The trick to doing this is to introduce variables for the coordinates for one of these points. Then this is the key idea: since we have two variables (i. e. , unknowns), we want to come up with two equations that these variables satisfy.
