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Question

Find the equations of the lines tangent to the graph of the function y = x² −2x + 1 such that each of the lines passes through the point P(2,−1).

Answer 1

Note: Two lines pass through (2, -1) that are tangent to the parabola. They form a crux "holding" the parabola whose vertex is at (1, 0).

As Michael A. correctly notes, the slope of the tangents is f'(x), or 2x - 2

Using point-slope form

y - (-1) = (2x - 2)(x - 2)

y + 1 = 2x² - 6x + 4

y = 2x² - 6x + 3, yet y = x² - 2x + 1

x² - 2x + 1 = 2x² - 6x + 3

0 = x² - 4x + 2

x = 2 ± √2

These are the x-coordinates of the points of tangency.

From here you can determine the equation of the two tangent lines.

 

Answer 2

Please disregard my previous solution to this problem. Mark M. is correct. In addition, this problem has already been addressed by Michael J. from NY previously. I apologize for any confusion. Thank you, Mark M.