A particle travels according to the equation `x=at^(3),y=bt^(3)`. The equation of the trajectory is

Options

A. `y=(ax^(2))/(b)`
B. `y=(bx^(2))/(a)`
C. `y=(bx)/(a)`
D. `y=(bx^(3))/(a)`

Answer

To find the equation of the trajectory for the given equations \( x = at^3 \) and \( y = bt^3 \), we will follow these steps:

### Step 1: Express \( t \) in terms of \( x \)
From the equation \( x = at^3 \), we can solve for \( t \):
\[ t^3 = \frac{x}{a} \]
Taking the cube root of both sides gives:
\[ t = \left(\frac{x}{a}\right)^{\frac{1}{3}} \]

### Step 2: Substitute \( t \) into the equation for \( y \)
Now, we substitute this expression for \( t \) into the equation for \( y \):
\[ y = bt^3 \]
Substituting \( t^3 \) from Step 1:
\[ y = b\left(\frac{x}{a}\right) \]

### Step 3: Simplify the equation
Now we can simplify this equation:
\[ y = \frac{b}{a} x \]

### Step 4: Write the final equation of the trajectory
The equation \( y = \frac{b}{a} x \) represents a straight line, which is the trajectory of the particle in the \( xy \)-plane.

Thus, the equation of the trajectory is:
\[ y = \frac{b}{a} x \]

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