If the roots of the quadratic equation $(a - b)x^2 + (b - c)x + (c - a) = 0$ are equal, prove that $2a = b + c$.
Step-by-Step Solution
Key Concept: Notice $x = 1$ is a root because sum of coefficients $(a-b) + (b-c) + (c-a) = 0$. Since roots are equal, both roots are $1$. Product of roots $= (c-a)/(a-b) = 1 \times 1 = 1 \Rightarrow c - a = a - b \Rightarrow 2a = b + c$.
Sum of coefficients $= (a - b) + (b - c) + (c - a) = 0 \Rightarrow x = 1$ is a root. [1.0 Mark]
Since roots are equal, both roots are $1$. [0.5 Mark]
Product of roots $= \dfrac{c - a}{a - b} = 1 \Rightarrow c - a = a - b \Rightarrow 2a = b + c$. Proved! [1.5 Marks]
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🎯 Official CBSE Marking Scheme:
Identifying $x = 1$ as a root: 1.0 Mark
Deducing both roots are $1$: 0.5 Mark
Equating product $(c-a)/(a-b) = 1 \Rightarrow 2a = b + c$: 1.5 Marks
Correct Answer: