Final answer:
To find the tangent lines parallel to x² = 2y - 2 on the curve y - x²/(x + 1), calculate the slope of the given line, find the derivative of the curve, equal it to the line's slope to find points of tangency, and use point-slope form to get the tangent equations.
Step-by-step explanation:
To find the equations of the tangent lines to the curve y - x²/(x + 1) that are parallel to the line x² = 2y - 2, we need to follow a specific sequence of steps:
- Determine the slope of the given line by rearranging it into slope-intercept form, which gives us y = (1/2)x² + 1. This line has a parabolic shape and its derivative will represent the slope of the tangent to any point on the parabola. Thus, the slope of this tangent line is the derivative of (1/2)x², which is x.
- Find the derivative of the given curve y - x²/(x + 1) to determine the slope of its tangent lines.
- Set the derivative of the curve equal to the slope obtained from step 1 (x) and solve for x to find the points where the tangents are parallel to the given line.
- Use these x-values in the original curve equation to find the corresponding y-values, identifying the points of tangency.
- Finally, use the point-slope form with the points of tangency and the common slope (x) to write the equations of the desired tangent lines.