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,
Create a free account to view solution
View Solution FreeMore Application of Derivative Questions
The equation of normal to the curve y = x3 − 2x2 + 4 at the point x = 2 is-...If at any point S of the curve by2 = (x + a)3, the relation between subnormal SN and subtangent ST be p(SN) = q(ST)2 the...If f(x) =where 0 < a < b < , then the equation f′(x) = 0 has, in the interval (a, b) -...The maximum area of the rectangle whose sides pass through the angular points of a given rectangle of sides a and b is...Water is dripping out from a conical funnel of semi-vertical angle at the uniform rate of 2 cm3/sec in its surface area ...