Hi all, I am struggling to finish this problem and was seeing if anyone could help me.
There are two circles.
Circle 1 has a center at (2,4), and radius 4.
Circle 2 has a center at (14,9) and radius 9.
I need to find the equation for common tangent line of these two circles. The circles touch so there is just one exterior tangent line.
Any help is appreciated, Thanks.
Hello,
you'll find all necessary methods here: http://www.mathhelpforum.com/math-he...712-post1.html
If you want to get the equation using calculus, follow these steps:
1- Write the circles as an equation. (Like ). Then, convert them to functions of x. The thing is, one equation will give you two functions of x, each of them making top and bottom semicircles. Take the functions of top semicircles. For example,
(top semicircle), (bottom semicircle)
Now apply it two your circles and take only the functions of top semicircles. Call them f(x), g(x)
2- Let's call the tangent points and .
3-
i. Slope of the line is
ii. Slope of the line is
iii. Slope of the line is
So,
First solve and find as a function of . Then plug this in and solve . You'll find . I think you can do the rest easily ^^
This calculations may be hard to do by hand. Using a plotter and a CAS will make your work easier. Good luck
P.S: If you can't solve it, write it here and I or someone else will probably help you wherever you're stuck.
In this situation, the problem is MUCH easier. Since the circles are externally tangent to each other, your tangent line is perpendicular to the segment connecting your centers, and passes through the point of the way across that segment.
Slope of segment:
Perpendicular slope:
Point that the tangent passes through:
I'll let you find the equation through that point with your perpendicular slope.
It seems that Henderson is trying to calculate the internal tangent, that is the one that passes through the point of contact of the two circles. But that’s not the tangent that is wanted, is it?
In any case, the internal tangent does not pass through . (It should pass through the point of contact of the circles, which is .)