A radius vector of point A relative to the origin varies with time t as `vec r = at hat i - bt^2 hat j` where `a and b` are constant. The equation of point's trajectory is.

Explanation

To find the equation of the trajectory of the point A 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 can be expressed in terms of its components:
- The \(x\) component is given by \(x = a t\).
- The \(y\) component is given by \(y = -b t^2\).

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

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

### Step 4: Rearrange the equation
To express the equation in a standard form, we can rearrange it:
\[ \frac{b}{a^2} x^2 + y = 0 \]
Or, multiplying through by \(a^2\):
\[ b x^2 + a^2 y = 0 \]

### Final Equation of the Trajectory
Thus, the equation of the trajectory of the point A is:
\[ a^2 y + b x^2 = 0 \]