Question

If the angle between the curves \( y = e^{(x+4)} \) and \( x^2 y = 1 \) at the point \( (1, 1) \) is \( \theta \), then \[ \sin \theta + \cos \theta = \dots \]

• A. \( \frac{7}{5} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{8}{7} \)
• D. \( \frac{6}{5} \)

Hint:

To find the angle between curves, first compute the derivatives of both curves at the given point and then apply the formula for the angle between two curves.

Answer 1

The Correct Option is A

The angle between two curves at a point is given by the formula: \[ \tan \theta = \left| \frac{f'(x_1) - g'(x_1)}{1 + f'(x_1)g'(x_1)} \right| \] where \( f'(x_1) \) and \( g'(x_1) \) are the derivatives of the curves at the point of intersection. Step 1: Find the derivatives of the curves \( y = e^{(x+4)} \) and \( x^2 y = 1 \) at \( x = 1 \). Step 2: Calculate the slope of the tangent lines of both curves at the point \( (1, 1) \). Step 3: Use the formula for the angle between two curves to compute \( \sin \theta + \cos \theta \), which results in \( \frac{7}{5} \). % Final Answer The value of \( \sin \theta + \cos \theta \) is \( \frac{7}{5} \).