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
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
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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).
Updated On: Jun 3, 2026
- 4a
- a
- 8a
- 2a
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The Correct Option is D
Solution and Explanation
Step 1: Concept
For any curve, the length of the sub-normal at any point is given by the formula $L_{sn} = \left|y \frac{dy}{dx}\right|$.
Step 2: Meaning
For the parabola $y^2 = 4ax$, differentiating both sides with respect to $x$ gives $2y \frac{dy}{dx} = 4a \implies y \frac{dy}{dx} = 2a$.
Step 3: Analysis
Notice that the sub-normal expression $\left|y \frac{dy}{dx}\right|$ simplifies directly to $|2a|$. This shows that for any standard parabola $y^2 = 4ax$, the length of the sub-normal is a constant value of $2a$, completely independent of the choice of coordinates of the point.
Step 4: Conclusion
Therefore, the given information about the length of the tangent being $4a\sqrt{5}$ is an extra parameter, and the sub-normal length is always $2a$, matching option (D).
Final Answer: (D)
For any curve, the length of the sub-normal at any point is given by the formula $L_{sn} = \left|y \frac{dy}{dx}\right|$.
Step 2: Meaning
For the parabola $y^2 = 4ax$, differentiating both sides with respect to $x$ gives $2y \frac{dy}{dx} = 4a \implies y \frac{dy}{dx} = 2a$.
Step 3: Analysis
Notice that the sub-normal expression $\left|y \frac{dy}{dx}\right|$ simplifies directly to $|2a|$. This shows that for any standard parabola $y^2 = 4ax$, the length of the sub-normal is a constant value of $2a$, completely independent of the choice of coordinates of the point.
Step 4: Conclusion
Therefore, the given information about the length of the tangent being $4a\sqrt{5}$ is an extra parameter, and the sub-normal length is always $2a$, matching option (D).
Final Answer: (D)
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