# Thread: Two tangent lines to a circle

1. ## Two tangent lines to a circle

Given two intersecting lines of the form y = mx+c, which are both tangential to a circle of radius r, how do I find an algebraic solution for:

• the location of the circle centre
• the location of each line/circle intersection

I guess there are two possible answers, as the circle could be in one of two positions.

Many thanks,
Siskin

2. Hint :

If r_1 and r_2 are the given lines, the center C of any circle tangential to r_1 and r_2 satisfies

d ( C , r_1 ) = d ( C , r_2 ) = r .

3. Give the line equations and radius.Without definition there are infinite solutions.

bjh

4. Originally Posted by siskin
Given two intersecting lines of the form y = mx+c, which are both tangential to a circle of radius r, how do I find an algebraic solution <--- we need the equations of the lines and the length of the radius to help you for:

• the location of the circle centre
• the location of each line/circle intersection

I guess there are two possible answers, as the circle could be in one of two <--- actually there are 4 possible positions positions.

Many thanks,
Siskin
Let $\displaystyle \alpha$ denote the acute angle between the 2 lines and e the distance between the point of intersection of the 2 lines and the center of a circle.

Then you know that

sin($\displaystyle \alpha$/2)= r/e

and

cos($\displaystyle \alpha$/2) = r/e

(see attachment)

5. I view this problem as a geometry problem with trig not allowed.A coordinate geometry solution could be described but it would seem that a geometry student would be given the required data and asked for answers.

bjh