Final answer:
To find the point where the normal to the curve y = x² - 2x + 3 is parallel to the y-axis, we need to determine the equation of the normal and find its point of intersection with the curve. The slope of the normal is the negative reciprocal of the slope of the tangent. The point where the normal is parallel to the y-axis is (1,2).
Step-by-step explanation:
To find the point where the normal to the curve y = x² - 2x + 3 is parallel to the y-axis, we need to determine the equation of the normal and find its point of intersection with the curve. The slope of the normal is the negative reciprocal of the slope of the tangent. The tangent can be found by taking the derivative of the equation. Differentiating y = x² - 2x + 3 gives us y' = 2x - 2. So, the slope of the tangent at any point (x,y) on the curve is 2x - 2. The slope of the normal is then -1 / (2x - 2).
A line parallel to the y-axis has a slope of infinity. Therefore, we need to find the point(s) where the slope of the normal is infinity. Setting -1 / (2x - 2) equal to infinity and solving for x, we have 2x - 2 = 0, which gives us x = 1. Substituting x = 1 back into the equation of the curve, we get y = 1² - 2(1) + 3 = 2. Therefore, the point where the normal to the curve is parallel to the y-axis is (1,2).
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