Geometry/Trigonometry
posted by Sam .
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

Geometry/Trigonometry 
Steve
Both circles are tangent to the xaxis, at (2,0) and (8,0).
The line joining their centers is inclined at an angle θ such that
tanθ = (91)/(82) = 4/3
So, the two tangent lines meet at an angle 2θ. Thus, our slope m of the other line is
tan2θ = (8/3)/(116/9) = 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
(x2)^2 + (y1)^2 = 1
(x2)^2 + (24/7 x+b1)^2 = 1
x^24x+4 + 576/49 x^2  48/7 (b1)x + (b1)^2 = 1
625/49 x^2  (48b20)/7 x + (b^22b+4) = 0
For that to have a single solution, the discriminant must be zero, so
((48b20)/7)^2  4(625/49)(b^22b+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+%28x2%29^2%2B%28y1%29^2%3D1+and+%28x8%29^2%2B%28y9%29^2%3D81%2C+y%3D24%2F7+x+%2B+30%2F7 
Geometry/Trigonometry 
Sam
Wow thanks a lot that helped!
Respond to this Question
Similar Questions

algebra
Rectangle R is formed by joining the centers of two congruent tangent circles and then drawing radii perpendicular to a common external tangent. If the perimeter of R is 60, what is its area? 
Trig
Three circles with radii of 4, 5, and 6 cm, respectively, are tangent to each other externally. Find the angles of the triangle whose vertexes are the centers of the circles. 
GEOMETRY(CIRCLE)
Three circles with different radii have their centers on a line. The two smaller circles are inside the largest circle, and each circle is tangent to the other two. The radius of the largest circle is 10 meters. Together the area of … 
trig
Two circles of radii 5 and 3 cm, respectively, intersect at two points. At either point of intersection, the tangent lines to the circles form a 60◦ angle, as in Figure 2.2.4 above. Find the distance between the centers of the … 
Trig/Precalc
Two questions that I would really appreciate some hints on: 1) Circles with centers (2,1) and (8,9) have radii 1 and 9, respectively. The equation for a common external tangent can be written in the form y=mx+b with 0<m. What is … 
GEOMETRY
Two circles with radii a and b touch each other externally . Let c be the radius of a circle which touches these two circles as well as their common tangent to the two circles . Then relate a,b and c through an equation 
math
Two circles have a radii of 15 and 95. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles? 
math
Two circles have a radii of 15 and 95. If the two external tangents to the circles intersect at 60 degrees, how far apart are the centers of the circles? 
Trigonometry
Two circles, whose radii are 12 inches and 16 inches respectively, intersect. The angle between the tangents at either of the points of intersection is 29'30'. Find the distance between the centers of the circles. 
geometry
Two overlapping identical circles have radii of 6.17. The area of their overlapping sections is 42.7. How far apart are the centers of the circles?