# Tangent Lines of Two Circles Intersect At Point

## Homework Statement

The tangent lines of two circles intersect at point (11/3,2/3). What are the two points that each tangent line touches on its respective circle?

## Homework Equations

Circle 1: x^2 + (y-3)^2 =5
Circle 2: (x-2)^2 + (y+3)^2 = 2

## The Attempt at a Solution

I found the derivatives of each circle.

Circle 1: y'(x) = -(x)/(y-3)
Circle 2: y'(x) = (2-x)/(y+5)

Do I have to use the slope-intercept equation somehow? y-yo=m(x-xo)

I'm not quite sure what do to next... :S

help anyone??

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

The tangent lines of two circles intersect at point (11/3,2/3). What are the two points that each tangent line touches on its respective circle?

## Homework Equations

Circle 1: x^2 + (y-3)^2 =5
Circle 2: (x-2)^2 + (y+3)^2 = 2

## The Attempt at a Solution

I found the derivatives of each circle.

Circle 1: y'(x) = -(x)/(y-3)
Circle 2: y'(x) = (2-x)/(y+5)

Do I have to use the slope-intercept equation somehow? y-yo=m(x-xo)

I'm not quite sure what do to next... :S
Hello VectorCereal. Welcome to PF !

Yes, using the slope intercept equation of a line can be helpful. In this case, (x0, y0) = (11/3, 2/3) .

Also, it looks to me like the problem can be solved for either circle independently of the other circle.

For Circle 1:

If you plug y'(x) = -(x)/(y-3) in for m in the slope intercept equation, you get the equation of another circle. See where this circle intersects with Circle 1.

Check your answer, because this seems like a weird method of solution !