Question

If the length of the tangent at a point on the parabola $y^{2}=4ax$ is $4a\sqrt{5}$, then the length of the sub-normal at that point is

• A. 4a
• B. a
• C. 8a
• D. 2a

Hint:

For the standard horizontal parabola $y^2 = 4ax$, the length of the sub-normal is constant at any point on the curve and is always equal to $2a$ (half of the latus rectum).

Answer 1

The Correct Option is D

Step 1: Recall the sub-normal length.
For any curve, the length of the sub-normal at a point is $\left|y\,\dfrac{dy}{dx}\right|$.
Step 2: Differentiate the parabola.
For $y^2=4ax$, differentiate both sides: $2y\,\dfrac{dy}{dx}=4a$, so $y\,\dfrac{dy}{dx}=2a$.
Step 3: Read off the sub-normal.
So $\left|y\,\dfrac{dy}{dx}\right|=|2a|=2a$.
Step 4: Notice it is constant.
This value $2a$ does not depend on which point we pick on the parabola. The sub-normal is the same everywhere.
Step 5: Handle the extra data.
The given tangent length $4a\sqrt5$ tells us which point we are at, but it does not change the sub-normal because that is always $2a$.
Step 6: Final value.
\[ \text{Sub-normal}=2a. \] \[ \boxed{2a} \]