\[\begin{array}{1 1}(1)-\frac{\pi}{4}&(2)\frac{\pi}{4}\\(3)0&(4)\frac{\pi}{2}\end{array}\]

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$x=e^t \cos t$

$y=e^t \sin t$

$\large\frac{dx}{dt} $$=e^{t} x - \sin t+ \cos t .e^t. \qquad \large\frac{dy}{dt}$$=e^{t}. \cos t + \sin t e^{t}$

$\large\frac{dx}{dt} $$=e^{t}(x -\cos t - \sin t) \qquad \large\frac{dy}{dt}$$=e^{t}. (\cos t + \sin t )$

$\large\frac{dy}{dx} =\large\frac{dy/dt}{dx/dt}=\frac{e^t (\cos t + \sin t)}{e^t (\cos t + \sin t)}$

$\large\frac{dy}{dx} =\large\frac{ \cos t + \sin t }{\cos t - \sin t}$

$=> \cos t + \sin t =0$

$\sin t = -\cos t$

$=> \large\frac{\sin t }{\cos t}$$=-1$

$\tan t=-1$

$t= -\large\frac{ \pi}{4}$

Hence 1 is the correct answer

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