Multiple choice

The roots of the equation $ \displaystyle \sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6} $ are

  1. $ \displaystyle \frac{3}{13},\frac{2}{13} $
  2. $ \displaystyle \frac{13}{6},\frac{11}{6} $
  3. $ \displaystyle \frac{9}{13},\frac{4}{13} $

  4. None of these
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C Correct answer
Explanation

Let y = sqrt(x/(1-x)). Then y + 1/y = 13/6. Solving 6y^2 - 13y + 6 = 0 gives y = 2/3 or 3/2. If sqrt(x/(1-x)) = 2/3, then x/(1-x) = 4/9, so 9x = 4 - 4x, 13x = 4, x = 4/13. If y = 3/2, x = 9/13.

AI explanation

Let y = sqrt(x/(1-x)), changing the equation to y + 1/y = 13/6. Multiplying by 6y gives the quadratic 6y^2 - 13y + 6 = 0, which factors to (3y-2)(2y-3) = 0 and gives y = 2/3 or y = 3/2. Squaring y = sqrt(x/(1-x)) results in y^2 = x/(1-x); substituting y = 2/3 gives 4/9 = x/(1-x), meaning 13x = 9 and x = 9/13. Substituting y = 3/2 gives 9/4 = x/(1-x), meaning 13x = 4 and x = 4/13. Therefore, the roots are 9/13 and 4/13.

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