Trigonometric EquationHard
Question
The normal to the curve x = a(cosθ + θ sinθ), y = a( sinθ - θ cosθ) at any point ′θ′ is such that
Options
A.it passes through the origin
B.it makes angle 
+ θ with the x-axis
+ θ with the x-axisC.it passes through 
D.it is at a constant distance from the origin
Solution
Clearly 
= tan θ ⇒ slope of normal = - cot θ
Equation of normal at ′θ′ is
y - a(sin θ - θ cos θ) = - cot θ(x - a(cos θ + θ sin θ)
⇒ y sin θ - a sin2 θ + a θ cos θ sin θ = - x cos θ + a cos2 θ + a θ sin θ cos θ
⇒ x cos θ + y sin θ = a
Clearly this is an equation of straight line which is at a constant distance ′a′ from origin.
= tan θ ⇒ slope of normal = - cot θEquation of normal at ′θ′ is
y - a(sin θ - θ cos θ) = - cot θ(x - a(cos θ + θ sin θ)
⇒ y sin θ - a sin2 θ + a θ cos θ sin θ = - x cos θ + a cos2 θ + a θ sin θ cos θ
⇒ x cos θ + y sin θ = a
Clearly this is an equation of straight line which is at a constant distance ′a′ from origin.
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