Question

ifx=a/b+b/a and y = a/b-b/a then let's show that x⁴y⁴-2x²y²=16​

Answer 1

Answer:

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Step-by-step explanation:

Answer 2

Question

Hello questioner. The question was corrected to

"If $x=\dfrac{a}{b} +\dfrac{b}{a}$ and $y=\dfrac{a}{b} -\dfrac{b}{a}$, show that $x^4-2x^2y^2+y^4=16$."

Required Knowledge

• $(x+y)(x-y)=x^2-y^2$

This identity is appropriate for this question, because when we see $x^4-2x^2y^2+y^4$, we can recognize it is the perfect square of $x^2-y^2$.

So, this question looks hard, but it is not harder than it seems. Let's solve this step-by-step.

Solution

We see that,

$\begin{cases} & x+y=\dfrac{2a}{b} \\ & x-y=\dfrac{2b}{a} \end{cases}$

The product of two is the key to the answer.

$\implies (x+y)(x-y)=4$

This implies that $x^2-y^2=4$. Now, on squaring both sides we obtain,

$\implies (x^2-y^2)^2=4^2$

$\implies x^4-2x^2y^2+y^4=16$

Therefore shown.