If the quadratic equation (b2 + c2) x2 - 2(a + b)cx + (c2 + a2) = 0 has equal roots, then
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a, b, c are in G.P
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a, b, c are in A.P
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a, c, b are in G.P
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a, c, b are in A.P
For a quadratic equation Ax^2 + Bx + C = 0 to have equal roots, the discriminant D = B^2 - 4AC must be zero. Substituting the given coefficients, we get 4(a+b)^2 c^2 - 4(b^2+c^2)(c^2+a^2) = 0, which simplifies to a^2 c^2 + b^2 c^2 - b^2 c^2 - b^2 a^2 - c^4 - c^2 a^2 = 0. Rearranging gives b^2 a^2 + c^4 = 0, which implies c^2 = ab, meaning a, c, b are in G.P.
For a quadratic equation to have equal roots, its discriminant must equal zero, so we set B^2 - 4AC = 0. Substituting the coefficients gives (-2(a + b)c)^2 - 4(b^2 + c^2)(c^2 + a^2) = 0, which simplifies to 4(a + b)^2c^2 - 4(b^2c^2 + b^2a^2 + c^4 + a^2c^2) = 0. Dividing by 4 and expanding the square yields a^2c^2 + 2abc^2 + b^2c^2 - b^2c^2 - a^2b^2 - c^4 - a^2c^2 = 0. Combining like terms leaves 2abc^2 - a^2b^2 - c^4 = 0, which factors as -(c^2 - ab)^2 = 0. Therefore, c^2 = ab, which is the exact condition for the three terms a, c, and b to form a geometric progression. The correct result is that a, c, b are in geometric progression.