1.

If x = k(θ + sin θ) and y = k (1 + cos θ), then what is the derivative of y with respect to x at θ = π/2?   1. -12. 03. 14. 2

Answer» Correct Answer - Option 1 : -1

Concept:

If x = f(θ), y =  f(θ) then, 

By chain Rule, we have 
\(\rm \frac {dy}{dx} = \dfrac {\frac {dy}{d\theta}}{\frac {dx}{d\theta}}\)

Calculations:

Given, x = k(θ + sin θ)

⇒\(\rm \dfrac {dx}{dθ} = k (1+cos \;θ)\)

y = k (1 + cos θ) 

⇒\(\rm \dfrac {dy}{dθ} = k (-sin \;θ)\)

By chain rule, we have

\(\rm \dfrac {dy}{dx} = \dfrac {\dfrac {dy}{d\theta}}{\dfrac {dx}{d\theta}}\)

⇒ \(\rm \dfrac{dy}{dx} = \dfrac {-ksin \;\theta}{k(1+ cos\;\theta)}\)

⇒ \(\rm \dfrac{dy}{dx} = \dfrac {-sin \;\theta}{(1+ cos\;\theta)}\)

Put θ = π/2

\(\rm \dfrac{dy}{dx} = \dfrac {-sin \;\dfrac {\pi}{2}}{(1+ cos\;\dfrac {\pi}{2})}\)

⇒ \(\rm \dfrac{dy}{dx} = -1\)

Hence, if x = k(θ + sin θ) and y = k (1 + cos θ), then the derivative of y with respect to x at θ = π/2 is -1.


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