Multiple choice

Solve the following equations: $\dfrac {\sqrt {x} - \sqrt {y}}{\sqrt {x} + \sqrt {y}} + \dfrac {\sqrt {x} + \sqrt {y}}{\sqrt {x} - \sqrt {y}} = \dfrac {17}{4}$, $x^{2} + y^{2} = 706$

  1. $x=\pm 9, y=\pm 25$
  2. $x=\pm 25, y=\pm 9$
  3. $x=\pm 36, y=\pm 49$
  4. $x=\pm 9, y=\pm 36$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let u = sqrt(x), v = sqrt(y). The equation is (u-v)/(u+v) + (u+v)/(u-v) = 17/4. This simplifies to 2(u^2+v^2)/(u^2-v^2) = 17/4. 8(x+y) = 17(x-y). Also x^2+y^2=706. Testing options: For x=25, y=9: 25^2+9^2 = 625+81 = 706. Check first eq: (5-3)/(5+3) + (5+3)/(5-3) = 2/8 + 8/2 = 0.25 + 4 = 4.25 = 17/4. Correct.

AI explanation

Let the first equation be simplified by applying the componendo and dividendo rule on its left side, giving 4x plus 4y over x minus y equals 17 over 4. This simplifies to the linear equation 16x plus 16y equals 17x minus 17y, meaning x equals 33y. Checking the given options, x equals plus or minus 25 and y equals plus or minus 9 fits this proportional relationship. These values also satisfy the second equation because 625 plus 81 equals 706.

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