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Question: If \(\frac{x^{2}}{(x^{2} + a^{2})(x^{2} + b^{2})} = k\left( \frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2...

If x2(x2+a2)(x2+b2)=k(a2x2+a2−b2x2+b2)\frac{x^{2}}{(x^{2} + a^{2})(x^{2} + b^{2})} = k\left( \frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}} \right)then k =

A

a2−b2a^{2} - b^{2}

B

1a+b\frac{1}{a + b}

C

1a−b\frac{1}{a - b}

D

1a2−b2\frac{1}{a^{2} - b^{2}}

Answer

1a2−b2\frac{1}{a^{2} - b^{2}}

Explanation

Solution

x2=k[a2(x2+b2)−b2(x2+a2)]x^{2} = k\lbrack a^{2}(x^{2} + b^{2}) - b^{2}(x^{2} + a^{2})\rbrack

⇒\Rightarrow x2=k[(a2−b2)x2]⇒1=k(a2−b2)x^{2} = k\lbrack(a^{2} - b^{2})x^{2}\rbrack \Rightarrow 1 = k(a^{2} - b^{2})

∴k=1a2−b2\therefore k = \frac{1}{a^{2} - b^{2}}.