Question

At which point is the slope of the curve \( y^2 = 4x \) equal to the slope of the line \( y = x + 3 \)?

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

Hint:

The slope of a line in the form \( Ax + By + C = 0 \) is \( -A/B \). The slope of a curve at a point is the value of its derivative at that point. To find a tangent parallel to a line, equate their slopes.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to find a point (x, y) on the curve \( y^2 = 4x \) where the slope of the tangent to the curve is the same as the slope of the given line. The slope of a curve at a point is given by its derivative \( \frac{dy}{dx} \) at that point.
Step 2: Key Formula or Approach:
1. Find the slope of the line \( y = x + 3 \).
2. Find the derivative \( \frac{dy}{dx} \) of the curve \( y^2 = 4x \) using implicit differentiation.
3. Set the slope of the curve equal to the slope of the line and solve for y.
4. Substitute the value of y back into the equation of the curve to find the corresponding value of x.
Step 3: Detailed Explanation or Calculation:
1. Slope of the line:
The line is given by \( y = 1x + 3 \), which is in the slope-intercept form \( y = mx + c \). The slope (m) is 1.
2. Slope of the curve:
Differentiate the curve's equation \( y^2 = 4x \) with respect to x:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4x) \] \[ 2y \frac{dy}{dx} = 4 \] \[ \frac{dy}{dx} = \frac{4}{2y} = \frac{2}{y} \] 3. Equate the slopes:
We set the slope of the curve equal to the slope of the line:
\[ \frac{2}{y} = 1 \] \[ y = 2 \] 4. Find the x-coordinate:
Now substitute \( y = 2 \) into the equation of the curve \( y^2 = 4x \):
\[ (2)^2 = 4x \] \[ 4 = 4x \] \[ x = 1 \] The required point is (1, 2). This point lies on the curve, and the slope of the tangent at this point is 1.
Step 4: Final Answer:
The point at which the slopes are equal is (1, 2).