If the roots of the equation are equal, then which one of the following is correct?
For the following two (02) items: Consider the equation abx^{2}+bcx+ca=cax^{2}+abx+bc
- A. ac=b^{2}
- B. a+c=2b
- C. \frac{1}{a}+\frac{1}{c}=\frac{1}{2b}
- D. \frac{1}{a}+\frac{1}{c}=\frac{2}{b} ✓
Correct Answer: D. \frac{1}{a}+\frac{1}{c}=\frac{2}{b}
Explanation
Rewrite the equation as (ab-ca)x^2 + (bc-ab)x + (ca-bc) = 0. The sum of the coefficients is (ab-ca) + (bc-ab) + (ca-bc) = 0, which means x=1 is a root. Since the roots are equal, both roots are 1. The product of the roots is \frac{ca-bc}{ab-ca} = 1. This simplifies to ca-bc = ab-ca \implies 2ca = b(a+c). Dividing by abc gives \frac{2}{b} = \frac{1}{a} + \frac{1}{c}.
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