The normal to the curve represented parametrically by x=a(cos⁡θ+θsin⁡θ) and y=a(sin⁡θ−θcos⁡θ)at any point θ, is such that it

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The normal to the curve represented parametrically by

$x = a (\cos θ + θ \sin θ) and y = a (\sin θ - θ \cos θ)$

at any point $θ$ , is such that it

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passes through the origin

makes a constant angle with the x-axis

does not touch a fixed circle

is at a constant distance from the origin

answer is B.

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Detailed Solution

$\frac{d y}{d x} = \tan θ$ , so equation the equation of normal is

$y - a (\sin θ - θ \cos θ) = - \cot θ (x - a (\cos θ + θ$

$\sin θ)$ which simplifies to $x \cos θ + y \sin θ = a .$

The distance of the normal from origin is $| a | .$

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