Application of DerivativeHard
Question
The normal to the curve x = a (cos θ + θ sin θ) y = a (sin θ - θ cos θ) any point ′θ′ is such that
Options
A.it makes a constant angle with the x-axis
B.it passes through the origin
C.it is at a constant distance from the origin
D.None of the above
Solution
Given, x = a (cos θ + θ sin θ)
and y = a (sin θ - θ cos θ)
∴
= a(- sin θ + sin θ + θ cos θ)
= a = θ cos θ
and
= a (cos θ - cos θ + θ sin θ)
= a θ sin θ ⇒ 
= tan θ
Thus, equation of normal is
⇒ - x cos θ + aθ sinθ cosθ + α cos2θ
= y sinθ + aθ sinθcosθ + αcos2 θ
⇒ = x cos θ + y sin θ = a
whose distance from origin is,
and y = a (sin θ - θ cos θ)
∴
= a(- sin θ + sin θ + θ cos θ) = a = θ cos θ
and
= a (cos θ - cos θ + θ sin θ) = a θ sin θ ⇒ = tan θ Thus, equation of normal is
⇒ - x cos θ + aθ sinθ cosθ + α cos2θ
= y sinθ + aθ sinθcosθ + αcos2 θ
⇒ = x cos θ + y sin θ = a
whose distance from origin is,
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