Question

A particle moves in $x-y$ plane according to rule $x = a\,\sin\omega t$ and $y = a \,\cos\omega t$. The particle follows

• A. an elliptical path
• B. a circular path
• C. a parabolic path
• D. a straight line path inclined equally to x and y-axes

Answer 1

The Correct Option is B

To determine the path of the particle as described by the equations x = a \sin\omega t and y = a \cos\omega t, let's analyze the given functions:

1. The equation x = a \sin\omega t represents the horizontal position of the particle as a function of time.
2. The equation y = a \cos\omega t represents the vertical position of the particle as a function of time.

Both \sin\omega t and \cos\omega t are trigonometric functions that vary between -1 and 1. To determine the path described by these parametric equations, we can eliminate the parameter \omega t as follows:

From x = a \sin\omega t, we have:

\sin\omega t = \frac{x}{a}

From y = a \cos\omega t, we have:

\cos\omega t = \frac{y}{a}

Using the Pythagorean identity \sin^2\theta + \cos^2\theta = 1, we can write:

\left(\frac{x}{a}\right)^2 + \left(\frac{y}{a}\right)^2 = (\sin\omega t)^2 + (\cos\omega t)^2 = 1

This simplifies to the equation:

\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1

Which is the standard form of the equation of a circle centered at the origin with radius a.

Therefore, the path of the particle is a circle. This confirms that the correct option is "a circular path".