Question

A tangent to the curve \( x = at^2,\; y = 2at \) is perpendicular to the X-axis. Then the point of contact is

• A. \( (0,-a) \)
• B. \( (0,0) \)
• C. \( (0,2a) \)
• D. \( (0,a) \)

Hint:

For parametric curves, a vertical tangent occurs when \( \frac{dx}{dt}=0 \).

Answer 1

The Correct Option is B

Step 1: Write the parametric equations.
\[ x = at^2,\qquad y = 2at \]

Step 2: Find the slope of the tangent.
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2a}{2at} = \frac{1}{t} \]

Step 3: Condition for perpendicular to X-axis.
A tangent perpendicular to the X-axis is vertical, so \[ \frac{dx}{dt} = 0 \Rightarrow 2at = 0 \Rightarrow t = 0 \]

Step 4: Find the point of contact.
At \( t=0 \): \[ x=0,\; y=0 \]

Step 5: Conclusion.
The point of contact is \[ \boxed{(0,0)} \]