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A circle is tangent to the x-axis (the line y = 0) and to the line 3x…
Question
A circle is tangent to the x-axis (the line y = 0) and to the line 3x - 4y = 0, lies below the line 3x - 4y = 0, and passes through the point (5,1). There are exactly two such circles. If the product of their radii equals m/n in lowest terms, find m+n.
✓ Verified answer: 35checked by our engine — not a guess
Step-by-step solution
Let the centre be (h,k) and radius r.
Tangent to y = 0 with the circle above it: k = r.
Tangent to 3x - 4y = 0, lying on the side where 3h - 4k > 0: |3h - 4k|/5 = r => (3h - 4k)/5 = r.
With k = r: 3h - 4r = 5r => 3h = 9r => h = 3r.
Passes through (5,1): (h-5)^2 + (k-1)^2 = r^2. Substitute h = 3r, k = r:
(3r-5)^2 + (r-1)^2 = r^2 => 9r^2 - 30r + 25 + r^2 - 2r + 1 = r^2 => 9r^2 - 32r + 26 = 0.
This quadratic in r has two positive roots r1, r2 (discriminant 1024 - 936 = 88 > 0). By Vieta, the product of the radii is r1 r2 = 26/9.
m/n = 26/9, m+n = 35.
Final answer35
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