How to find common tangents for two circles? - Analytic geometry

[MayaMaya121]

Hi,
English isn't my native language so please excuse me if I make grammar mistakes.

So as my math homework I received a problem that asked me to find all of the common tangents of two circles. When I tried to solve the problem I found out that I missed an answer, which was a equation of the line : X=4. I found out that was because I used the general equation of a line y=mx+b (I understand that I missed the answer because the line X=4 doesn't have a natural slope.) and I made two equations that the distance from the middle point of the circle to the tangent was the radius. And after a lot of calculations and thinking I received 3 answers instead of four, and I made another equation to find the X=4 answer manually.
That kept me thinking, is there another way to find the common tangents of two circles without missing the X=something answer?
And please try to explain it as simple as you can because I am only an 8th grader...

 

[Deleted member 4993]

Hi,
English isn't my native language so please excuse me if I make grammar mistakes.

So as my math homework I received a problem that asked me to find all of the common tangents of two circles. When I tried to solve the problem I found out that I missed an answer, which was a equation of the line : X=4. I found out that was because I used the general equation of a line y=mx+b (I understand that I missed the answer because the line X=4 doesn't have a natural slope.) and I made two equations that the distance from the middle point of the circle to the tangent was the radius. And after a lot of calculations and thinking I received 3 answers instead of four, and I made another equation to find the X=4 answer manually.
That kept me thinking, is there another way to find the common tangents of two circles without missing the X=something answer?
And please try to explain it as simple as you can because I am only an 8th grader...

Please post the equations for the "three" tangents that you did find and associated work done to find those.

 

[MayaMaya121]

I don't have the homework since I am waiting for my teacher to check it. Is it enough if I will post the equations of the circles and the common tangents and explain how I found them?

 

[Deleted member 4993]

I don't have the homework since I am waiting for my teacher to check it. Is it enough if I will post the equations of the circles and the common tangents and explain how I found them?

Yes..... That would be justfine.

 

[MayaMaya121]

The circles: [MATH]\left(x-1\right)^2+\left(y-4\right)^2=1\:,\:\:\:\left(x-5\right)^2+\left(y-6\right)^2=9[/MATH]Their common tangents: [MATH]4x-3y+13=0\:,\:\:\:3x+4y-24=0\:\:,\:\:y=3\:\:\:,\:\:x=2[/MATH]
Since the radius is perpendicular to the tangent, I used two equations of the distance from the center of the circle to a general linear equation [MATH]y=mx+b[/MATH] (which I changed to [MATH]mx-y+b=0[/MATH]), and the distance from the center of the circle to the tangents is equal to the radius (I did an equation for each circle).