Question

A particle is moving in xy plane whose X and Y coordinates are x= 3 sin 2 Pi t and y=2 cos 2 Pi t where x and y are in metres and t is in seconds a) the path of particle is elliptical
b)the path of particle is circular
c)the path of particle is parabolic
d) the path of particle is straight line.

Answer 1

Answer:

The path is elliptical.

Explanation:

The general equation of ellipse is

x^2/a^2 + y^2/b^2 = 1.

Here we have x^2/9+y^2/4= 1.

the parametric equation for this ellipse satisfied the given data.

Answer 2

Answer:

The correct option for the path of the particle is option a) the path of the particle is elliptical.

Explanation:

To find the path of the particle, we have to eliminate the variable 't' and construct a relation between 'x' and 'y'. To do this follow the steps below.

Step 1:

Given:

                                       $x=3sin(2\pi t)\\y=2cos(2\pi t)$

we're going to use the trigonometric identity

                                     $sin^{2}(\theta)+cos^{2}(\theta)=1$

Step 2:

From the given equations for 'x' and 'y' we have,

                                       $\frac{x}{3}=sin(2\pi t)\\ \frac{y}{2}=cos(2\pi t)$

Squaring both sides in the above two equations we get,

                                      $(\frac{x}{3})^{2}=sin^{2}(2\pi t) \\ (\frac{y}{2})^{2}=cos^{2}(2\pi t)$

Step 3:

Adding the above two equations we get,

                                    $(\frac{x}{3})^{2}+(\frac{y}{2})^{2}=sin^{2}(2\pi t) +cos^{2}(2\pi t)$

Using the identity mentioned above the RHS of the above equation becomes 1.

                                    $(\frac{x}{3})^{2}+(\frac{y}{2})^{2}=1$

Step 4:

The above equation is an equation of an ellipse, with a semi-major axis of length 3/2 units and a semi-minor axis length of 2/2 units.

Therefore, the path of the particle is an ellipse with a semi-major axis of length 1.5 units and a semi-minor axis of length 1 unit.