Circles with centers (2,1) and (8,9) have radii 1 and 9, respectively. The equation of a common external tangent to the circles can be written in the form y=mx+b with m < 0. What is b?

I drew the diagram and the tangents but to no avail. I cannot seem to find the points where the tangent hits both circles because if I do, the problem would be solved. Help is appreciated, thanks in advance

Both circles are tangent to the x-axis, at (2,0) and (8,0).

The line joining their centers is inclined at an angle θ such that

tanθ = (9-1)/(8-2) = 4/3

So, the two tangent lines meet at an angle 2θ. Thus, our slope m of the other line is
tan2θ = (8/3)/(1-16/9) = -24/7

Now we have a line

y = -24/7 x + b

which must be tangent to both circles. That is, if we look for where the line intersects the circle, there must be a single solution.

Taking the first circle, we need

(x-2)^2 + (y-1)^2 = 1
(x-2)^2 + (-24/7 x+b-1)^2 = 1
x^2-4x+4 + 576/49 x^2 - 48/7 (b-1)x + (b-1)^2 = 1
625/49 x^2 - (48b-20)/7 x + (b^2-2b+4) = 0

For that to have a single solution, the discriminant must be zero, so

((48b-20)/7)^2 - 4(625/49)(b^2-2b+4) = 0

b = 30/7 and 80/7

we want the smaller value. The larger one will be on the other side of the circle. So,

y = -24/7 x + 30/7

To see the graphs, visit

http://www.wolframalpha.com/input/?i=plot+%28x-2%29^2%2B%28y-1%29^2%3D1+and+%28x-8%29^2%2B%28y-9%29^2%3D81%2C+y%3D-24%2F7+x+%2B+30%2F7

Wow thanks a lot that helped!

To find the equation of a common external tangent to the circles, you can follow these steps:

Step 1: Draw the two circles with their given centers and radii. In this case, one circle has the center at (2,1) and a radius of 1, while the other circle has the center at (8,9) and a radius of 9.

Step 2: Connect the two centers of the circles to create a line segment.

Step 3: Find the midpoint of the line segment created in Step 2. To do this, use the Midpoint Formula:
- The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints:
(2 + 8)/2 = 10/2 = 5.
- The y-coordinate of the midpoint is the average of the y-coordinates of the two endpoints:
(1 + 9)/2 = 10/2 = 5.
So, the midpoint of the line segment is (5, 5).

Step 4: Draw a line through the midpoint of the line segment so that it passes through the centers of the circles. This line is the line connecting the two centers and it represents the line of centers.

Step 5: Find the slope of the line of centers. To do this, use the slope formula:
- The slope of a line passing through (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1).
Using the coordinates of the two centers, the slope of the line of centers is:
m = (9 - 1) / (8 - 2) = 8 / 6 = 4/3.

Step 6: Since we need the slope of the tangent line (m < 0), take the negative reciprocal of the slope of the line of centers. In this case, the negative reciprocal of 4/3 is -3/4.

Step 7: Find the equation of the line passing through the midpoint with a slope of -3/4. To do this, use the point-slope form of a linear equation:
- The equation of a line passing through a point (x1, y1) with slope m is given by:
y - y1 = m(x - x1).
Plugging in the midpoint coordinates (5, 5) and the slope -3/4, we get:
y - 5 = -3/4(x - 5).

Step 8: Convert the equation to the slope-intercept form (y = mx + b) to find the value of b.
Distributing and rearranging the equation, we have:
y - 5 = -3/4x + 15/4.
Simplifying, we get:
y = -3/4x + 35/4.

Comparing this equation to the form y = mx + b, we see that the value of b is 35/4.

Therefore, the value of b is 35/4.