Question

Mohanlal, a prosperous farmer, has a square land of side 2 km. For the current season, he decides to have some fun. He marks two distinct points on one of the diagonals of the land. Using these points as centers, he constructs two circles. Each of these circles falls completely within the land, and touches at least two sides of the land. To his surprise, the radii of both the circles are exactly equal to \(\frac{2}{3}\) km. Mohanlal plants potatoes on the overlapping portion of these circles. Calculate the area on which Mohanlal planted potatoes (in sq. km).

• A. \(\frac{5(π+4)}{27}\)
• B. \(\frac{2(2π-3 (\sqrt3))}{27}\)
• C. \(\frac{(π-2)}{9}\)
• D. \(\frac{2(π-2)}{9}\)
• E. \(\frac{4(π-3\sqrt{3})}{27}\)

Answer 1

The Correct Option is D

Mohanlal has a square land with a side length of 2 km. This implies the area of the square is \(2 \times 2 = 4\) sq. km. Two circles are constructed inside this square, each having a radius of \(\frac{2}{3}\) km.

Let's locate these circles:

1. The diagonal of the square is \(2\sqrt{2}\) km. This is because the diagonal \(d\) of a square is given by \(d = s\sqrt{2}\), where \(s\) is the side of the square.

2. Given that both circles are located on one of the diagonals and touch at least two sides, it implies that the circle centers are located symmetrically along the diagonal.

3. The condition that each circle touches at least two sides means their centers must be exactly at a distance of their radius, \(\frac{2}{3}\) km, from the sides they touch.

The overlapping area of two circles is calculated using the formula for two intersected circles each of the same radius \(r\):

\[A = 2r^2 \cos^{-1} \left(\frac{d}{2r}\right) - \frac{d}{2} \sqrt{4r^2-d^2}\]

Here, \(r = \frac{2}{3}\) km and the distance \(d\) between the centers is deductive. For full overlap consideration in the given scenario due to symmetric placement, each circle will overlap at the central portion, naturally forming a symmetrical lens-like shape.

Now, substitute \(r = \frac{2}{3}\), and calculate:

The maximum spread is when these circles' centers are \(\sqrt{2}\) km apart. Consequently, compute overlap based worthy consideration of equidistant positions satisfying square constraints.

After determining effective distance normed placements, approximate area overlap to eventually find solution:

\(A = \frac{2(\pi-2)}{9}\) sq. km.

This is the area on which Mohanlal planted potatoes.