GATE CS 2020 QGA-8 — General Aptitude

To find the unpainted area in the annular space, we subtract the total area of the $n$ painted circles from the total area of the annular ring.

Step 1: Area of the annular ring
The outer radius of the annular ring is $b$ and the inner radius is $a$.
$$\text{Area of the annular space} = \pi b^2 - \pi a^2 = \pi(b^2 - a^2)$$

Step 2: Area of a single painted circle
Each painted circle touches both the inner boundary (radius $a$) and the outer boundary (radius $b$) of the annular space. Therefore, the diameter $d$ of each circle is the width of the annular space:
$$d = b - a$$ The radius $r$ of each painted circle is:
$$r = \frac{b - a}{2}$$ The area of one painted circle is:
$$\text{Area}_{\text{circle}} = \pi r^2 = \pi \left(\frac{b - a}{2}\right)^2 = \frac{\pi}{4}(b - a)^2$$

Step 3: Total painted area
Since there are $n$ such circles, the total painted area is:
$$\text{Total Painted Area} = n \times \frac{\pi}{4}(b - a)^2 = \frac{n\pi}{4}(b - a)^2$$

Step 4: Unpainted area
Subtracting the painted area from the total annular space area:
$$\text{Unpainted Area} = \pi(b^2 - a^2) - \frac{n\pi}{4}(b - a)^2$$ $$\text{Unpainted Area} = \pi \left[ (b^2 - a^2) - \frac{n}{4}(b - a)^2 \right]$$

This matches Option A.