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$

(1){\text{ }}\dfrac{\pi }{2} \\

(2){\text{ }}5\dfrac{\pi }{2} \\

(3){\text{ }}3\pi \\

(4){\text{ }}\dfrac{\pi }{4} \\

$

Answer
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Given- Circle 1: ${x^2} + {y^2} = 1$ and Circle 2: ${x^2} + {y^2} = 4$

Pair of lines: $\sqrt 3 ({x^2} + {y^2}) = 4xy$

Find the individual equations of the straight lines. We have

$\sqrt 3 {x^2} - 4xy + \sqrt 3 {y^2} = 0$

Factorizing,$\sqrt 3 {x^2} - 3xy - xy + \sqrt 3 {y^2} = 0$

Simplifying,$(x - \sqrt 3 y)(\sqrt 3 x - y)$

Line 1: $x - \sqrt 3 y = 0$ Line 2: $\sqrt 3 x - y = 0$

We need to find the shaded area. For this, we will use the sector formula.

When area of only a sector of a circle covering $\theta $ angle is to be found:

$\dfrac{\theta }{{2\pi }} \times \pi {r^2}$

Here, since we have to find the area of the region between two circles:$\dfrac{\theta }{{2\pi }} \times \pi [r_2^2 - r_1^2]$

The angle $\theta $ here is (60 - 30 = 30) i.e. $\dfrac{\pi }{6}$

${r_2} = 2{\text{ , }}{{\text{r}}_1} = 1$

On solving, we get

Area = $\dfrac{\pi }{4}$ .

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