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The figure below shows an annular ring with outer and inner as $b$ and $a$, respectively. The annular space has been painted in the form of blue colour circles touching the outer and inner periphery of annular space. If maximum $n$ number of circles can be painted, then the unpainted area available in annular space is _____.

- $\pi [(b^{2}-a^{2})-\frac{n}{4}(b-a)^{2}]$
- $\pi [(b^{2}-a^{2})-n(b-a)^{2}]$
- $\pi [(b^{2}-a^{2})+\frac{n}{4}(b-a)^{2}]$
- $\pi [(b^{2}-a^{2})+n(b-a)^{2}]$

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Best answer

Answer is Option (A)

We need to find the area of the ring first.

**Area of Ring $=$ Area of Outer Circle $-$ Area of Inner Circle **

$\qquad \qquad \quad =\pi b^2 - \pi a^2 = \pi (b^2-a^2)$

Now From this ring area we have to subtract the small **'$n$'** circles

**Area of $n$ Circles $ = (\pi d^2/4) \ast n = (\pi (b-a)^2 /4)\ast n$**

**The Required Unpainted Area $=$** **Ring Area $-$** **Area of Small '$n$' such Circles**

$\qquad \qquad \quad =\pi (b^2 - a^2) - (\pi d^2/4)*n$

$\qquad \qquad \quad = \pi [(b^2-a^2)- n/4(b-a)^2 ]$