A partical moves in the `x-y` plane according to the scheme `x=-8 sin pi t` and `y=-2 cos 2pi t`, where `t` is time. Find the equation of path of the particle

Options

A. `y=-2+(x^(2))/16`
B. `y^(2)=-2+(x^(2))/16`
C. `x^(2)=-2+(y^(2))/16`
D. `x=-2+(y^(2))/16`

Explanation

To find the equation of the path of the particle moving in the x-y plane according to the equations \( x = -8 \sin(\pi t) \) and \( y = -2 \cos(2\pi t) \), we can follow these steps:

### Step 1: Express \( \sin(\pi t) \) in terms of \( x \)
From the equation for \( x \):
\[ x = -8 \sin(\pi t) \]
We can solve for \( \sin(\pi t) \):
\[ \sin(\pi t) = -\frac{x}{8} \]

### Step 2: Use the double angle identity for cosine
The equation for \( y \) is:
\[ y = -2 \cos(2\pi t) \]
Using the double angle identity for cosine, we know:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Setting \( \theta = \pi t \), we have:
\[ \cos(2\pi t) = 1 - 2\sin^2(\pi t) \]
Substituting \( \sin(\pi t) \) from Step 1:
\[ \cos(2\pi t) = 1 - 2\left(-\frac{x}{8}\right)^2 \]
Calculating \( \left(-\frac{x}{8}\right)^2 \):
\[ \left(-\frac{x}{8}\right)^2 = \frac{x^2}{64} \]
Thus,
\[ \cos(2\pi t) = 1 - 2 \cdot \frac{x^2}{64} = 1 - \frac{x^2}{32} \]

### Step 3: Substitute back into the equation for \( y \)
Now substitute \( \cos(2\pi t) \) back into the equation for \( y \):
\[ y = -2 \left(1 - \frac{x^2}{32}\right) \]
Distributing the \(-2\):
\[ y = -2 + \frac{2x^2}{32} = -2 + \frac{x^2}{16} \]

### Step 4: Rearranging the equation
The final equation of the path of the particle can be written as:
\[ y = \frac{x^2}{16} - 2 \]

### Final Result
The equation of the path of the particle is:
\[ y = \frac{x^2}{16} - 2 \]