$a^2(b^2 - c^2)x^2 + b^2 (c^2 - a^2)x + c^2 (a^2 – b^2) = 0$
are equal $(a^2 \neq b^2 \neq c^2)$.
To determine the correct statement, we need to analyze the condition mentioned: 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, where \(a^2 \neq b^2 \neq c^2\).
When the roots of a quadratic equation are equal, the discriminant of the equation is zero. The quadratic equation is generally expressed in the format \(Ax^2 + Bx + C = 0\), where \(A\), \(B\), and \(C\) are coefficients. The discriminant \(\Delta\) is given by \(B^2 - 4AC\).
For our given quadratic equation:
- Coefficient \(A = a^2(b^2 - c^2)\)
- Coefficient \(B = b^2(c^2 - a^2)\)
- Coefficient \(C = c^2(a^2 - b^2)\)
Setting the discriminant to zero for equal roots:
\(B^2 - 4AC = (b^2(c^2 - a^2))^2 - 4(a^2(b^2 - c^2))(c^2(a^2 - b^2)) = 0\)
Expanding and simplifying this equation gives:
\(b^4(c^2 - a^2)^2 = 4a^2c^2(b^2 - c^2)(a^2 - b^2)\)
Solving this equation helps us understand the relationship between \(a^2, b^2,\) and \(c^2\).
When expanded and simplified, it reveals that \(a^2, b^2,\) and \(c^2\) satisfy the relationship of being in Harmonic Progression (HP).
Therefore, the correct option is: \(a^2, b^2, c^2\) are in HP.
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