Question : If $x^{4} + y^{4} + x^{2} y^{2} = 17 \frac{1}{16}$ and $x^{2} - x y + y^{2} = 5 \frac{1}{4}$ , then one of the values of $(x - y)$ is:
Option 1: $\frac{5}{2}$
Option 2: $\frac{3}{4}$
Option 3: $\frac{5}{4}$
Option 4: $\frac{3}{2}$

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Question : If $x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$ , then the value of $\frac{x^{2} + x y + y^{2}}{x^{2} - x y + y^{2}}$ is:

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Answer 1

Correct Answer: $\frac{5}{2}$

Solution : Given: $x^{4} + y^{4} + x^{2} y^{2} = 17 \frac{1}{16}$ ..........................(1)
$x^{2} - x y + y^{2} = 5 \frac{1}{4}$ ..........................(2)
⇒ $x^{4} + y^{4} + x^{2} y^{2} = (x^{2} - x y + y^{2}) (x^{2} + x y + y^{2})$
⇒ $17 \frac{1}{16} = 5 \frac{1}{4} (x^{2} + x y + y^{2})$
⇒ $\frac{273}{16} = \frac{21}{4} (x^{2} + x y + y^{2})$
⇒ $x^{2} + x y + y^{2} = \frac{13}{4}$ ..........................(3)
On adding (2) and (3), we get
$x^{2} + y^{2} = \frac{17}{4}$ ..................................(4)
Subtract (2) form (3), we get
$x y = - 1$ .....................................(5)
Now, Using the formula $(a - b)^{2} = a^{2} + b^{2} - 2 a b$
$(x - y)^{2} = x^{2} + y^{2} - 2 x y$
From (4) and (5), we get
⇒ $(x - y)^{2} = \frac{17}{4} - 2 \times (- 1)$
⇒ $(x - y)^{2} = \frac{17}{4} + 2 = \frac{25}{4}$
⇒ $(x - y) = \frac{5}{2}$
Hence, the correct answer is $\frac{5}{2}$ .

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Question : If $x^{4} + y^{4} + x^{2} y^{2} = 17 \frac{1}{16}$ and $x^{2} - x y + y^{2} = 5 \frac{1}{4}$ , then one of the values of $(x - y)$ is:
Option 1: $\frac{5}{2}$
Option 2: $\frac{3}{4}$
Option 3: $\frac{5}{4}$
Option 4: $\frac{3}{2}$