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If the equations ax^(2) +2bx +c = 0 an...

If the equations `ax^(2) +2bx +c = 0 and Ax^(2) +2Bx+C=0` have a common root and a,b,c are in G.P prove that `a/A , b/B, c/C ` are in H.P

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To prove that \( \frac{a}{A}, \frac{b}{B}, \frac{c}{C} \) are in Harmonic Progression (H.P) given that the equations \( ax^2 + 2bx + c = 0 \) and \( Ax^2 + 2Bx + C = 0 \) have a common root and \( a, b, c \) are in Geometric Progression (G.P), we can follow these steps: ### Step 1: Understand the condition of G.P Since \( a, b, c \) are in G.P, we have: \[ b^2 = ac \]
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