Question

find the asymptotes of the curve r theta cos theta = a cos 2theta ​

Answer 1

Answer:

Asymptotes are the lines which touch the curve at infinity.

Putting u=1r, then

u=θcosθacos2θ=F(θ)

When r→∞, u→0, or

θcosθacos2θθcosθθ=0=0=0,±π2

i.e. when r→∞, θ→0,±π2. So,

θ1=0,±π2

Differentiating F(θ) w.r.t θ,

F′(θ)F′(θ)=acos2θddθ(θcosθ)−θcosθddθ(acos2θ)a2cos22θ=acos2θ(cosθ−θsinθ)+2aθcosθsin2θa2cos22θ

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The equation of the asymptote in case of polar curves is given by,

rsin(θ−θ1)=1F′(θ1)

So in our case, asymptotes are,enter image description here