Which one of the following statements is correct?
Direction: Consider the following for the two (02) items that follow :<br>The roots of the quadratic equation a^{2}(b^{2}-c^{2})x^{2}+b^{2}(c^{2}-a^{2})x+c^{2}(a^{2}-b^{2})=0 are equal (a^{2} \neq b^{2} \neq c^{2})
- A. a^{2}, b^{2}, c^{2} are in AP.
- B. a^{2}, b^{2}, c^{2} are in GP.
- C. a^{2}, b^{2}, c^{2} are in HP. ✓
- D. a^{2}, b^{2}, c^{2} are neither in AP nor in GP nor in HP.
Correct Answer: C. a^{2}, b^{2}, c^{2} are in HP.
Explanation
The sum of the coefficients is a^2(b^2-c^2) + b^2(c^2-a^2) + c^2(a^2-b^2) = 0, indicating x=1 is a root. Since the roots are equal, both roots are 1. The product of the roots is \frac{c^2(a^2-b^2)}{a^2(b^2-c^2)} = 1. Rearranging yields c^2a^2 - c^2b^2 = a^2b^2 - a^2c^2 \implies 2a^2c^2 = b^2(a^2+c^2). Dividing by a^2b^2c^2 gives \frac{2}{b^2} = \frac{1}{a^2} + \frac{1}{c^2}, which means a^2, b^2, c^2 are in HP.
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