Question

(b) At which point is the slope of the line \( y = x + 1 \) equal to the slope of the curve \( y^2 = 4x \)?

• A. \( (1,2) \)
• B. \( (2,1) \)
• C. \( (1,-2) \)
• D. \( (-1,2) \)

Hint:

For curves, differentiate implicitly if \( y \) is squared or appears in complex forms.

Answer 1

The Correct Option is C

Step 1: Find slope of the given line \( y = x + 1 \). Since the equation is in slope-intercept form \( y = mx + c \), the slope is \[ m = 1. \] Step 2: Differentiate the curve equation \( y^2 = 4x \). Using implicit differentiation: \[ 2y \frac{dy}{dx} = 4 \quad \Rightarrow \quad \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y}. \] Step 3: Solve for \( y \) when \( \frac{dy}{dx} = 1 \). \[ 1 = \frac{2}{y} \Rightarrow y = 2. \] Plugging \( y = 2 \) into \( y^2 = 4x \), \[ 4 = 4x \Rightarrow x = 1. \] Thus, the point is \( (1,-2) \).