If tangent to the curve `x = at^2 ,y = 2at` is perpendicular to x-axis then its point of contact is

Options

A. (a.a)
B. (0,a)
C. (a,0)
D. (0,0)

Answer

To solve the problem, we need to find the point of contact on the curve defined by the parametric equations \( x = at^2 \) and \( y = 2at \), where the tangent to the curve is perpendicular to the x-axis.

### Step-by-Step Solution:

1. **Understanding the Condition**:
A tangent that is perpendicular to the x-axis has an undefined slope, which implies that the slope of the tangent line is vertical. This means that the derivative \( \frac{dy}{dx} \) at the point of contact must be infinite.

2. **Finding the Derivatives**:
We need to find \( \frac{dy}{dx} \) using the parametric equations:
\[ x = at^2 \quad \text{and} \quad y = 2at \]
To find \( \frac{dy}{dx} \), we use the chain rule:
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]
First, we calculate \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \):
\[ \frac{dy}{dt} = 2a \quad \text{and} \quad \frac{dx}{dt} = 2at \]
Therefore,
\[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \]

3. **Setting the Condition for Perpendicularity**:
For the tangent to be vertical, \( \frac{dy}{dx} \) must be undefined, which occurs when \( t = 0 \) (since \( \frac{1}{t} \) becomes undefined).

4. **Finding the Point of Contact**:
Now, substituting \( t = 0 \) back into the parametric equations to find the coordinates of the point of contact:
\[ x = a(0)^2 = 0 \quad \text{and} \quad y = 2a(0) = 0 \]
Thus, the point of contact is \( (0, 0) \).

5. **Conclusion**:
The point of contact where the tangent to the curve is perpendicular to the x-axis is:
\[ (0, 0) \]