Question

Two coincident lines are of equal length. One of them is moved clockwise with respect to the other such that they share a common end point until they coincide again. Let A be the area swept by the line and let B be the area swept when this line makes any angle $\theta (\theta <360)$ with fixed line. Then A:B is:

• A. 360 : $\theta$
• B. $\theta$ : 360
• C. $\theta$ : $\pi$
• D. $\pi$ : $\theta$

Answer 1

The correct option is A

360 : $\theta$

According to the initial position, the lines are coincident. Let the length of the line be r.

Then, one of the lines is rotated until they coincide again. This means the moving line makes ${360}^{\circ }$ with the fixed line which necessarily forms a circle.

So, the area swept by the line for ${360}^{\circ }$ = Area of the circle = $\pi \times r^2$ -----------(1)

When the moving line makes $\theta (\theta <360°)$ with fixed line, area swept will be equal to $\frac{\theta }{360}$ times the area of the circle

So, B = $\frac{\theta }{360}\times \pi \times r^2$ --------(2)

From (1) and (2),

The required ratio = A : B = 360 : $\theta$