$a^2(b^2 - c^2)x^2 + b^2 (c^2 - a^2)x + c^2 (a^2 – b^2) = 0$
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.
Similar Questions
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The quadratic equation 3x 2- (k 2+ 5k)x + 3k 2- 5k = 0 has real roots of equal magnitude and opposite sign. Which one of the following is correct?
Under what condition does the above equation have real roots?
Under what condition is \(\frac{1}{{{{\rm{\alpha }}^2}}} + \frac{1}{{{{\rm{\beta }}^2}}} < 1?\)
If both the roots of the equation x 2– 2kx + k 2- 4 = 0 lie between – 3 and 5, then which one of the following is correct?
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The roots α and β of a quadratic equation, satisfy the relations α + β = α 2+ β 2and αβ = α 2β 2. What is the number of such quadratic equations?
- Consider the following statements :
I. \(\sqrt{x} + x + 1 = 0\) has two irrational roots.
II. \(5\sqrt{x} - x - 4 = 0\) has two rational roots.
Which of the statements given above is/are correct ? Consider the following
1. α + β + αβ > 0
2. α 2β + β 2α > 0
Which of the above is/are correct?Consider the following:
1. β < - α
2. β < |α|
Which of the above is/are correct?
Important Questions from Quadratic Equations
- Find the zeros of the function \(f(x) = (x - 17)(x - 7)\)
- Find the roots of the following equation
$x^2 + 12x - 36 = 0$ - Which of the following equations are quadratic equations?
I) $x^2 + 4x + 23 = x + \frac{33}{28}$
II) $\frac{2}{3}x^2 + 6x + 2 = x^2 + 8$
III) $6x^2 + 4x + 18 = 6x^2 - 6x$ - Find the roots of the following equation:
$4x^2 + 4x - 3 = 0$ - Which of the following is a quadratic equation?
