Multiple choice

The value of $m$ for which the equation $\dfrac { a }{ x+a+m } +\dfrac { b }{ x+b+m } =1$ has roots equal in magnitude but opposite in sign is

  1. $\dfrac{a+b}{a-b}$
  2. $0$
  3. $\dfrac{a-b}{a+b}$
  4. $\dfrac2({a-b}){a+b}$.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Combining the fractions and simplifying gives the quadratic x^2 + 2mx + m^2 - ab = 0. Its roots have equal magnitude and opposite signs only when their sum is zero, so -2m = 0 and m = 0.

AI explanation

Multiplying the equation a/(x+a+m) + b/(x+b+m) = 1 by the common denominator yields a(x+b+m) + b(x+a+m) = (x+a+m)(x+b+m). Expanding both sides gives x(a+b) + 2ab + m(a+b) = x^2 + x(a+b+2m) + (a+m)(b+m). For the roots to be equal in magnitude but opposite in sign, their sum must be zero, meaning the coefficient of x must equal zero. Setting the coefficients equal gives a+b = a+b+2m, which simplifies to 2m = 0, so m = 0.

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