Final answer:
The values of h and k are found by using the slope of the normal line parallel to the given line and the point at which the curve passes. After finding two equations from the derivative and the point, we can solve for h and k. The solution yields h = -1 and k = 1.
Step-by-step explanation:
To find the values of h and k for the curve y = hx² + kx at the point (-1,2), where the normal is parallel to the line y - 2x + 4 = 0, we need to use the concept of derivatives to find the slope of the tangent and then the slope of the normal to the curve at (-1,2).
- The slope (m) of the line y - 2x + 4 = 0 is 2 because it's in the form y = mx + b; hence, m = 2.
- The slope of the normal is the negative reciprocal of the slope of the tangent to the curve at a given point. Thus, the slope of the normal is -1/2.
- We calculate the derivative of y = hx² + kx to find the slope of the tangent at the point (-1,2). The derivative is dy/dx = 2hx + k.
- Substituting x = -1 into the derivative gives us the slope at that point, which must be -2 (since it's the negative reciprocal of the normal's slope).
- So, 2h(-1) + k = -2, which simplifies to -2h + k = -2.
- We also know that the curve passes through the point (-1,2), so by substituting x = -1 and y = 2 into the curve equation, we get 2 = h(-1)² + k(-1), which simplifies to h - k = 2.
- Now, we have two equations: -2h + k = -2 and h - k = 2. Solving these simultaneously gives us h = -1 and k = 1.
Therefore, the correct answer is h = -1 and k = 1 (option d).