Question

Find the equation of the normal to a parabola which is perpendicular to a given line. This involves:

• A. Slope comparison
• B. Differentiation
• C. Both A and B
• D. None of these

Hint:

The slope of a normal to $y^2 = 4ax$ is often represented as $m$. The equation in slope form is $y = mx - 2am - am^3$. If you know the slope from the "given line," you can plug it directly into this formula!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
A normal to a curve at a point is a line perpendicular to the tangent at that point. To find its equation under a specific geometric constraint (perpendicular to another line), we need to determine the slope and the point of contact.

Step 2: Detailed Explanation
1. Differentiation (B): We differentiate the equation of the parabola ($y^2 = 4ax$) to find the slope of the tangent ($m_t = dy/dx$). The slope of the normal is then $m_n = -1/m_t$. 2. Slope Comparison (A): The problem states the normal is perpendicular to a "given line." If the given line has slope $m_L$, then the slope of our normal must be $m_n = -1/m_L$. We compare this required slope to the derivative-based slope to find the specific point on the parabola. 3. Synthesis: You cannot solve the problem without finding the derivative (to relate the point to the slope) and comparing slopes (to apply the perpendicularity condition).

Step 3: Final Answer
The process requires both slope comparison and differentiation.