Points where tangent line touches 2 circles

[madgab89]

Homework Statement

On the circles y^2+x^2=1 and y^2+(x-3)^2=4
There is a line with positive slope that is tangent to both circles. Determine the points at which this tangent touches each circle.

Homework Equations

the derivative of the first circle i found:
y'=-x/y

the derivative of the second cirlce I found:
y'=-2x+6/2y

and also
x^2+y^2=1

The Attempt at a Solution

so I have these equations for slopes:
-x₁/y₁

-2x₂+6/2y₂

y₂-y₁/x₂-x₁

Now where do I go from here, can someone get me started with the rearranging or whatever?

 

[rock.freak667]

What I think you should do is put the tangent as y=mx+c, then solve this with y^2+x^2=1, and when you simplify it into the form ax^2+bx+c=0, b^2-4ac=0 since it is a tangent. Then do the same with y^2+(x-3)^2=4 and you will get two equations in m and c.

 

[Dick]

Homework Statement

On the circles y^2+x^2=1 and y^2+(x-3)^2=4
There is a line with positive slope that is tangent to both circles. Determine the points at which this tangent touches each circle.

Homework Equations

the derivative of the first circle i found:
y'=-x/y

the derivative of the second cirlce I found:
y'=-2x+6/2y

and also
x^2+y^2=1

The Attempt at a Solution

so I have these equations for slopes:
-x₁/y₁

-2x₂+6/2y₂

y₂-y₁/x₂-x₁

Now where do I go from here, can someone get me started with the rearranging or whatever?

I think the main problem is that you haven't written down any equations yet. Those are expressions, not equations. Put in some equal signs. Do you want to say that all of those are equal to the unknown slope m?