A ardius vector of point A relative to the origin varies with time t as `vec(r)= at hat(j)-bt^(2) hat(j)` where a and b are constants. Find the equation of point's trajectory.

Explanation

To find the equation of the point's trajectory given the radius vector \(\vec{r} = a t \hat{i} - b t^2 \hat{j}\), we will follow these steps:

### Step 1: Identify the components of the radius vector
The radius vector is given as:
\[ \vec{r} = a t \hat{i} - b t^2 \hat{j} \]
From this, we can identify the x and y components:
- \(x = a t\)
- \(y = -b t^2\)

### Step 2: Solve for time \(t\) in terms of \(x\)
From the equation \(x = a t\), we can express \(t\) as:
\[ t = \frac{x}{a} \]

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

### Step 4: Simplify the equation for \(y\)
This simplifies to:
\[ y = -b \frac{x^2}{a^2} \]
or rearranging gives us:
\[ y = -\frac{b}{a^2} x^2 \]

### Conclusion
The equation of the point's trajectory is:
\[ y = -\frac{b}{a^2} x^2 \]

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