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Question

Prove that the equation r2cosθarcos2θ2a2cosθ=0 represents a straight line and a circle.

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Solution

r2cosθarcos2θ2a2cosθ=0r2cosθar(2cos2θ1)2a2cosθ=0r2cosθ2arcos2θ+ar2a2cosθ=0rcosθ(r2acosθ)+a(r2acosθ)=0(rcosθ+a)(r2acosθ)=0rcosθ+a=0,r2acosθ=0

Converting rcosθ+a=0 to Cartesian coordinates

x+a=0

which represents a straight line

Also r2acosθ=0

Multiplying by r on both sides

r22arcosθ=0(x2+y2)22ax=0x2+y22ax=0

which represents a circle .

Hence proved that the given equation represents a circle and a straight line


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