# Two tangent lines to a circle

• Apr 19th 2011, 08:15 AM
siskin
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
• Apr 19th 2011, 08:52 AM
FernandoRevilla
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 .
• Apr 19th 2011, 08:57 AM
bjhopper
Give the line equations and radius.Without definition there are infinite solutions.

bjh
• Apr 19th 2011, 11:13 AM
earboth
Quote:

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

and

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

(see attachment)
• Apr 19th 2011, 11:43 AM
bjhopper
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