Find 4th Tangent for 2 Circles Coordinate Geometry

[Michael_Light]

Homework Statement

2 circles have the equation x²+y²-2x-2y+1=0 and x²+y²-12x-12y+36=0 respectively. Both circle touches the x-axis, y-axis and the line 3x + 4y = 12. Find the fourth tangent of the 2 circles.

Homework Equations

The Attempt at a Solution

This is second part of the question, I solved the first part which require to find the equation of the 2 circles... And now i stuck at this final part... can anyone help me?

 

[ehild]

The line 3x+4y is not tangent to the circles. Show the original problem, please.

ehild

 

[Michael_Light]

Its 3x + 4y = 12... not 3x+4y=0...

 

[ehild]

Well, I wanted to say 3x+4y=12. Check if it is tangent to the circles.

ehild

 

[micromass]

Well, I wanted to say 3x+4y=12. Check if it is tangent to the circles.

ehild

I find that it is tangent to the circles. Perhaps you miscalculated??

But it's a good starter question to the OP: how do you check whether a line is tangent to the circle??

 

[Michael_Light]

If a line is tangent to the circle, then the perpendicular distance from the center of the circle to the given line is equals to the radius of the circle...

So what is the trick to find fourth tangent of the 2 circles? I cannot figure out how to find it..

 

[ehild]

The common tangent line has one common point with both circles. So the equation of a circle and that of the line have a single solution.
Draw those circles. The symmetry of the figure gives you hint about the other tangent.

ehild

 

[ehild]

I find that it is tangent to the circles. Perhaps you miscalculated??

Thank you, micromass!
You are right, my calculator played tricks with me. It IS a tangent line.

ehild

 

[SammyS]

The common tangent line has one common point with both circles. So the equation of a circle and that of the line have a single solution.
Draw those circles. The symmetry of the figure gives you hint about the other tangent.

ehild

Thank you, micromass!
You are right, my calculator played tricks with me. It IS a tangent line.

ehild

Graphing the two circles may lead one to think that the circles share a point of tangency. I did just that using Wolfram Alpha. I then included the given line, 3x + 4y = 12, and zoomed-in.

 

[ehild]

Nice pictures SammyS!

A pair of tangent lines to two circles are mirror images of each other to the line that connects the centre of the circles.
The centres of the circles (x-1)²+(y-1)²=1 and (x-6)²+(y-6)²=36 are (1,1) and (6,6); both lie on the y=x line. So a graph of the circles and the tangent lines is invariant when x and y are exchanged. With the change x<=>y in the equation of a tangent line, you get the other one.

ehild