Statement 1 − a , b , c are positive numbers and the chord of contact of the tangents drawn from any point on the circle x 2 + y 2 = a 2 to the circle x 2 + y 2 = b 2 touches the circle x 2 + y 2 = c

Question

Statement 1 − a , b , c are positive numbers and the chord of contact of the tangents drawn from any point on the circle x 2 + y 2 = a 2 to the circle x 2 + y 2 = b 2 touches the circle x 2 + y 2 = c 2 . Then a − 2 bk + c k 2 is non-negative for every k ∈ R . Statement 2 - The given circles are concentric.

TestHub · Accepted Answer

The chord of contact from ( a cos θ , a sin θ ) is x cos θ + y sin θ = a b 2 ​ . It touches x 2 + y 2 = c 2 , so a b 2 ​ = c , i.e. b 2 = a c . Then c k 2 − 2 bk + a ≥ 0 for all k since c > 0 and the discriminant 4 b 2 − 4 a c = 0 . Statement 1 is true. The circles are concentric, so Statement 2 is true, but it does not explain Statement 1.

Mathematics - Circle Question with Solution | TestHub

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