# For what value(s) of m is the line y=mx+6 tangent to the circle x²+y²=9?

##### 1 Answer

#### Explanation:

I highly recommend you visualize the equations on a graph - it is very helpful to do so. Imagine the function,

An interesting property of tangent lines is that a line drawn from the point of intersection to the circle's center is *always* perpendicular to the tangent line.

A way to approach this problem is to exploit this property of perpendicular tangency. Now, originally, I wanted to solve this problem using purely just the equations, however, it became really complex.

However, I realized that this can be done with a bit of simple geometry and a touch of trigonometry, here's how:

Refer to the above diagram. The line segments

Here's what we know:

Let's start:

I use degrees instead of radians here.

Using:

where

Since slopes are

Therefore the values of

**But wait there's more**

Remember how I said there was an algebraic way to do it? And that it's overly complicated? Here it is (no diagrams):

Any point on this circle can be expressed in polar coordinates:

Therefore any point on the circle can be rewritten as:

A tangent line on the circle is perpendicular to the radius drawn from the point of intersection. Let

Then we need to write a function describing the tangent line:

where

We know that

We plug it in to find

Thus

This will give us a tangent line to the circle. However, we have restraints, and that is

So solve for

We plug it back into

Told you it was overly complex.