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For the curve given parametrically by x = a(cos θ + θ sin θ) and y = a(sin θ - θ cos θ), the normal drawn at a point corresponding to parameter θ has which property?
- It always goes through the origin.
- It forms an angle of π/2 + θ with the positive x-axis.
- It passes through the point (aπ/2, -a).
- Its distance from the origin remains fixed.
Correct answer: Its distance from the origin remains fixed.
Solution
The normal line at a point on the curve maintains a constant distance from the origin due to the specific parametric equations, which define a relationship between x and y that ensures the normal's position relative to the origin does not change as θ varies.
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