Mathematics
A and B are centres of circles of radii 9 cm and 2 cm such that AB = 17 cm and C is the centre of the circle of radius r cm which touches the above circles externally. If ∠ACB = 90°, write an equation in r and solve it.
Circles
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Answer
From figure,
In △ABC,
By pythagoras theorem,
⇒ AB2 = AC2 + BC2
⇒ 172 = (r + 9)2 + (r + 2)2
⇒ 289 = r2 + 81 + 18r + r2 + 4 + 4r
⇒ 289 = 2r2 + 85 + 22r
⇒ 2r2 + 22r + 85 - 289 = 0
⇒ 2r2 + 22r - 204 = 0
⇒ 2(r2 + 11r - 102) = 0
⇒ r2 + 11r - 102 = 0
⇒ r2 + 17r - 6r - 102 = 0
⇒ r(r + 17) - 6(r + 17) = 0
⇒ (r - 6)(r + 17) = 0
⇒ r - 6 = 0 or r + 17 = 0
⇒ r = 6 or r = -17.
Since, radius cannot be negative.
⇒ r = 6 cm.
Hence, equation is r2 + 11r - 102 = 0 and r = 6 cm.
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