What is the equation of the normal line to the curve y = x^2 - 4x + 5 at the point (1, 2)?
Answer and explanation
Correct answer: C. x - 2y + 3 = 0
- Step 1: Find the tangent slope by differentiating: y' = 2x - 4.
- Step 2: Evaluate at x = 1: m_tan = 2(1) - 4 = -2.
- Step 3: The slope of the normal line is the negative reciprocal: m_norm = -1 / (-2) = 1/2.
- Step 4: Write point-slope equation: y - 2 = (1/2)(x - 1) => 2y - 4 = x - 1 => x - 2y + 3 = 0.
Answer
x - 2y + 3 = 0.
Why not the others: '2x + y - 4 = 0' is the equation of the tangent line (y - 2 = -2(x - 1)); 'x + 2y - 5 = 0' negates the normal slope; '2x - y = 0' uses the tangent slope with wrong intercept.
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