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Which one of the following statements is incorrect?
- The sub-tangent to the curve x²y² = 16a⁴ at the point (-2a, 2a) has length 2a.
- The line x + y = 3 is a normal to the curve x²y = 4y.
- The curves y = -4x² and y = e^x/2 intersect orthogonally.
- If a lies in (-1, 0), then the tangent at every point on the curve y = (2/3)x³ - 2ax² + 2x + 5 makes an acute angle with the positive x-axis.
Correct answer: The curves y = -4x² and y = e^x/2 intersect orthogonally.
Solution
The curves y=-4x^2 and y=e^(x/2) cannot intersect: the first is always <=0 while the second is always >0, so they have no common point and cannot intersect orthogonally. That statement is therefore incorrect, while the sub-tangent (length 2a), the normal x+y=3 to x^2=4y, and the acute-angle tangent claim are all correct.
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