If we call the two circles C1 and C2, respectively, then note that (2,1) lies on C2. So, the external tangents will be bisected by the line joining the two centers.
Note that both circles are tangent to the x-axis.
So, figure the slope of the line joining the two centers, and then the other tangent has twice that slope.
I'm thinking on the other one for a bit.
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 b?
2) In triangle ABC, AB=BC, and BD is an altitude. Point E is an extension of AC such that BE=10. The values of tan(CBE), tan(DBE), and tan(ABE) form a geometric progression, and the values of cot(DBE), cot(CBE), and cot(DBC) form an arithmetic progression. What is the area of triangle ABC?
Thank you so much in advance for any help!
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Okay, thanks so much for your hints on the first one! I got it using a combination of point-slope form and arc tangents. I'll wait to hear more on #2 if you have the time :)
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