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Question

If $$x(a-b+\frac{ab}{a-b})=y(a+b-\frac{ab}{a+b}) and x+y=2a^3$$ ,   then what is x-y equal to ?

  1. $$-2b^3$$
  2. $$-2ab^3$$
  3. $$2b^3$$
  4. $$2ab^3$$
Answer

‘.’  $$  x(a-b+\frac{ab}{a-b})=y(a+b-\frac{ab}{a+b})$$

   $$  \frac{x}{y}= \frac{(a+b-\frac{ab}{a+b})}{(a-b+\frac{ab}{a-b})}$$

     By, Componendo & Dividendo –

  $$   \frac{x+y}{x-y} = \frac{(a+b-\frac{ab}{a+b})+(a-b+\frac{ab}{a-b})}{(a+b-\frac{ab}{a+b})-(a-b+\frac{ab}{a-b})}$$

 $$     \frac{x+y}{x-y} = \frac{(2a + \frac{ab}{a-b} – \frac{ab}{a+b})}{(2b – \frac{ab}{a+b}- \frac{ab}{a-b})}$$

 $$     \frac{ x+y}{x-y} = \frac{(2a + \frac{a^2b+ab^2 – (a^2b-ab^2)}{a^2-b^2})}{(2b – \frac{(a^2b-ab^2 + a^2b+ab^2)}{ a^2-b^2})}$$

 $$     \frac{x+y}{x-y} = \frac{(2a + \frac{2ab^2}{a^2-b^2})}{2b – \frac{2a^2b}{a^2-b^2})}$$

$$    \frac{ x+y}{x-y} = \frac{(\frac{2a^3 – 2ab^2 +2ab^2}{a^2-b^2})}{(\frac{2a^2b – 2b^3 – 2a^2b}{a^2-b^2})}$$

$$   \frac{x+y}{x-y} = \frac{2a^3}{-2b^3}     $$

      $$ So, x-y= -2b^3$$

Answer (A)