Question

The equation of normal to the curve \( y = \sin \left( \frac{\pi x}{4} \right) \) at the point (2, 5) is

• A. \( x + y = 5 \)
• B. \( y = 5 \)
• C. \( x = 2 \)
• D. \( x + y = 2 \)

Hint:

The equation of the normal to a curve is found by first determining the slope of the tangent and then using the negative reciprocal for the normal slope. If the tangent slope is 0, the normal is vertical.

Answer 1

The Correct Option is C

Step 1: Find the derivative of the curve.
To find the equation of the normal line, we first compute the derivative of \( y = \sin \left( \frac{\pi x}{4} \right) \) to obtain the slope of the tangent at the point \( (2, 5) \). The derivative is \( \frac{dy}{dx} = \frac{\pi}{4} \cos \left( \frac{\pi x}{4} \right) \). At \( x = 2 \), the slope of the tangent is 0.

Step 2: Find the slope of the normal.
The slope of the normal line is the negative reciprocal of the tangent slope. Since the tangent slope is 0, the normal slope is undefined, which implies that the normal line is a vertical line. Thus, the equation of the normal is \( x = 2 \).

Step 3: Conclusion.
Thus, the equation of the normal is \( x = 2 \), corresponding to option (C).