Find the value of dy/dx at θ = pi/4, if x = aeθ(sin θ – cos θ)

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Question:

Find the value of \dfrac{dy}{dx} at \theta = \dfrac{\pi}{4}, if x = ae^{\theta}(\sin \theta - \cos \theta) and y = ae^{\theta}(\sin \theta + \cos \theta).          ……..   [CBSC 2008, 2014]

Solution:

Given, x = ae^{\theta}(\sin \theta - \cos \theta) and y = ae^{\theta}(\sin \theta + \cos \theta)

Taking, x = ae^{\theta}(\sin \theta - \cos \theta)

Differentiating with respect to θ, we get

\dfrac{dx}{d\theta} = ae^{\theta}(\cos \theta + \sin \theta)+a(\sin \theta - \cos \theta).e^{\theta}

= ae^{\theta}(\cos \theta + \sin \theta +\sin \theta - \cos \theta)

= 2ae^{\theta} \sin \theta}

Again taking, y = ae^{\theta}(\sin \theta + \cos \theta)

Differentiating with respect to θ, we get

\dfrac{dy}{d\theta}=ae^{\theta}(\cos \theta - \sin \theta)+a(\sin \theta + \cos \theta).e^{\theta}

=ae^{\theta}(\cos \theta - \sin \theta + \sin \theta + \cos \theta)

= 2ae^{\theta} \cos \theta

\therefore \dfrac{dy}{dx}= \dfrac{dy/d\theta}{dx/d\theta}

=\dfrac{2ae^{\theta} \cos \theta}{2ae^{\theta} \cos \theta}

\Rightarrow \dfrac{dy}{dx}= \cot \theta

\Rightarrow \left[\dfrac{dy}{dx}\right]_{\theta = \frac{\pi}{4}} = \cot \frac{\pi}{4} = 1

Some other question:

Q 1: If \tan^{-1}(\frac{y}{x}) = \log \sqrt{x^2+y^2}, Prove that \dfrac{dy}{dx} = \dfrac{x + y}{x - y}.    ……..[CBSC 2020]

Solution : For solution click here

Q 2: If y^x = e^{y-x}, then prove that \dfrac{dy}{dx}=\dfrac{(1+\log y)^2}{\log y}   …….[CBSC 2013]

Solution: For solution click here

Q 3:If \cos y = x \cos(a + y), with \cos a \neq \pm 1, then prove that \dfrac{dy}{dx}=\dfrac{\cos^2(a + y)}{\sin a}. Hence show that \sin a \dfrac{d^2y}{dx^2}+ \sin 2(a + y) \dfrac{dy}{dx}=0. ……..      [CBSC  2016]

Solution: For solution click here

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