Question

Let \(y = f(x)\) be any curve on the X-Y plane and \(P\) be a point on the curve. Let \(C\) be a fixed point not on the curve. The length \(PC\) is either a maximum or a minimum. Then:

• A. \(PC\) is perpendicular to the tangent at \(P\)
• B. \(PC\) is parallel to the tangent at \(P\)
• C. \(PC\) meets the tangent at an angle of \(45^\circ\)
• D. \(PC\) meets the tangent at an angle of \(60^\circ\)

Hint:

When dealing with optimization problems involving distances and curves, the distance is maximized or minimized when the vector from the point of interest is either parallel or perpendicular to the tangent.

Answer 1

The Correct Option is A

Step 1: The length \( PC \) is the distance from the fixed point \( C \) to a point \( P \) on the curve. We know \( PC \) is either a maximum or a minimum.

Step 2: To find a maximum or minimum for length \( PC \), the line connecting \( P \) and \( C \) must be perpendicular to the tangent at \( P \).

Step 3: The vector \( \overrightarrow{PC} \) is perpendicular to the tangent line at \( P \) when the distance is at its minimum or maximum.

Answer: (A) \( PC \) is perpendicular to the tangent at \( P \)