# Show that the normal at any point to the curve x = acosθ + aθsinθ, y = asinθ – aθcosθ is at a constant distance from the origin.

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Show that the normal at any point to the curve x = acosθ + aθsinθ, y = asinθ – aθcosθ is at a constant distance from the origin.

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Given,

x = a cosθ – a θ sinθ

y = a sinθ - a θ cosθ

Distance from origin (0,0) to (i),

$= \begin{vmatrix} \frac{0.cos\,+0.sin\,\theta-a}{\sqrt{cos^2\theta+sin^2\theta}} \\[0.3em] \end{vmatrix}=a$