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Verified Answer :

syed514

r=e^θ
using polar coordinates
x=rcos(θ)=e^θcosθ
y=rsin(θ)=e^θsinθ

Horizontal tangents occur where dy/dx=0 and vertical tangents occur where

dy/dx→∞⟹dx→0.

dy/dx=dy/dθ/dx/dθ

dy/dx=e^θcosθ+e^θsinθ/−e^θsinθ+e^θcosθ

So horizontal tangents occur where
e^θcosθ+e^θsinθ=0⟹θ=−π/4

Vertical tangents occur where
−e^θsinθ+e^θcosθ=0⟹θ=π/4

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