Question

Find angle of Intersection between curves r=a(1-cos thetha) and r=2acos thetha​

Answer 1

Answer:

The curves meet where they have the same r and θ . At that point tan(2θ)=1 , so 2θ=π4+nπ and θ=π8+nπ2 . where n is an arbitrary integer. So there are four solutions with θ between 0 and 2π .

We need to find the angle between the curves at each of these points. To avoid confusion we should use separate symbols for the coordinates for the two curves, so let’s use subscripts 1 and 2 .

For the first curve

dr1dθ1=−2asin(2θ1)

and for the second curve

dr2dθ2=2acos(2θ2) .

At the points of intersection θ1=θ2=θ , say, and dr1dr2=−tan(2θ) . And when θ=π8+nπ2 , dr1dr2=−1 . Imagine you are at a point of intersection looking along the second curve. The first curve would appear at a gradient −1 , that is an angle π4 to your direction.

So the angle between the curves is π4 .

Step-by-step explanation:

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