Question:
If the roots of the equation \( \frac{\alpha}{x-\alpha} + \frac{\beta}{x-\beta} = 1 \) are equal in magnitude but opposite in sign, then \( \alpha + \beta \) is equal to
If the roots of the equation \( \frac{\alpha}{x-\alpha} + \frac{\beta}{x-\beta} = 1 \) are equal in magnitude but opposite in sign, then \( \alpha + \beta \) is equal to
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For equations with roots equal in magnitude but opposite in sign, check the sum of the roots and use symmetry to solve for parameters.
Updated On: Apr 22, 2026
- 0
- 1
- 2
- None of these
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The Correct Option is A
Solution and Explanation
Step 1: Start with the equation.
Given the equation: \[ \frac{\alpha}{x - \alpha} + \frac{\beta}{x - \beta} = 1 \] Let's find a common denominator: \[ \frac{\alpha(x - \beta) + \beta(x - \alpha)}{(x - \alpha)(x - \beta)} = 1 \] Simplifying the numerator: \[ \frac{(\alpha + \beta)x - 2\alpha\beta}{(x - \alpha)(x - \beta)} = 1 \]
Step 2: Solve for the expression.
Now, equate the numerator to the denominator: \[ (\alpha + \beta)x - 2\alpha\beta = (x - \alpha)(x - \beta) \] Expanding the right-hand side: \[ (\alpha + \beta)x - 2\alpha\beta = x^2 - (\alpha + \beta)x + \alpha\beta \]
Step 3: Compare coefficients.
Comparing the coefficients of \( x \) and the constant terms: 1. \( \alpha + \beta = -(\alpha + \beta) \) (gives us \( \alpha + \beta = 0 \)). 2. \( -2\alpha\beta = \alpha\beta \) (this condition is automatically satisfied).
Step 4: Conclusion.
Thus, \( \alpha + \beta = 0 \), corresponding to option (A).
Given the equation: \[ \frac{\alpha}{x - \alpha} + \frac{\beta}{x - \beta} = 1 \] Let's find a common denominator: \[ \frac{\alpha(x - \beta) + \beta(x - \alpha)}{(x - \alpha)(x - \beta)} = 1 \] Simplifying the numerator: \[ \frac{(\alpha + \beta)x - 2\alpha\beta}{(x - \alpha)(x - \beta)} = 1 \]
Step 2: Solve for the expression.
Now, equate the numerator to the denominator: \[ (\alpha + \beta)x - 2\alpha\beta = (x - \alpha)(x - \beta) \] Expanding the right-hand side: \[ (\alpha + \beta)x - 2\alpha\beta = x^2 - (\alpha + \beta)x + \alpha\beta \]
Step 3: Compare coefficients.
Comparing the coefficients of \( x \) and the constant terms: 1. \( \alpha + \beta = -(\alpha + \beta) \) (gives us \( \alpha + \beta = 0 \)). 2. \( -2\alpha\beta = \alpha\beta \) (this condition is automatically satisfied).
Step 4: Conclusion.
Thus, \( \alpha + \beta = 0 \), corresponding to option (A).
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