For a quadratic equation to have equal (repeated) roots, its discriminant must be zero.\nIf the roots of a(b - c)x^2 + b(c - a)x + c(a - b) = 0 are equal, which relation between a, b and c must be true?

Introduction / Context:
This question tests the condition for equal roots of a quadratic equation. For a general quadratic Ax^2 + Bx + C = 0, the roots are equal (a repeated root) when the discriminant is zero: B^2 - 4AC = 0. Here A, B, and C are expressions in a, b, and c, so we compute the discriminant symbolically and then simplify to obtain the required relationship among parameters.

Given Data / Assumptions:

• Quadratic: a(b - c)x^2 + b(c - a)x + c(a - b) = 0
• A = a(b - c), B = b(c - a), C = c(a - b)
• Equal roots condition: B^2 - 4AC = 0
• Assume denominators used in relations are non-zero where required.

Concept / Approach:
Compute discriminant D = B^2 - 4AC using A, B, C above. Then factor the expression to identify the parameter constraint. After simplifying, the condition reduces neatly to b(a + c) = 2ac, which can be rearranged into the option form 2/b = 1/a + 1/c.

Step-by-Step Solution:
Identify coefficients: A = a(b - c), B = b(c - a), C = c(a - b) Equal roots require: D = B^2 - 4*A*C = 0 Compute: B^2 = b^2(c - a)^2 Compute: 4AC = 4 * a(b - c) * c(a - b) After algebraic simplification, D becomes a perfect square: D = (ab - 2ac + bc)^2 For D = 0, we need ab - 2ac + bc = 0 Factor b from the first and third terms: b(a + c) - 2ac = 0 So b(a + c) = 2ac Divide both sides by abc (assuming a, b, c non-zero for this step): (a + c)/(ac) = 2/b But (a + c)/(ac) = 1/a + 1/c Therefore: 2/b = 1/a + 1/c

Verification / Alternative check:
If 2/b = 1/a + 1/c holds, then b = 2ac/(a + c). Substituting into the discriminant form (ab - 2ac + bc)^2 makes the inside zero, so the discriminant is zero and the roots are equal. This confirms the necessity and sufficiency of the relation.

Why Other Options Are Wrong:
a + b + c = 0 does not generally force the discriminant to be zero for these coefficients.
a b c = a b + b c + c a is a different symmetric condition and does not match the derived discriminant factor.
2 b = (1/a) + (1/c) has unit/structure mismatch and is not equivalent to b = 2ac/(a + c).
b = (a + c)/(a c) equals 1/a + 1/c, but the required relation is 2/b equals that quantity, so it misses the factor 2.

Common Pitfalls:
Using the wrong equal-roots condition (some confuse it with B = 0), or expanding A, B, C incorrectly due to sign errors like mixing (c - a) with (a - c). Another common mistake is failing to factor the discriminant properly and missing that it is a perfect square.

Final Answer:
2/b = (1/a) + (1/c)