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Question

Find the length of the perpendicular drawn from the origin on the tangent to the curve x=a(θsinθ),y=a(1cosθ) at θ=π2.

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Solution

Given curve is x=a(θsinθ),y=a(1cosθ)
At θ=π2,x=a(π/2sinπ/2)=a2(π2),y=a(1cosπ/2)=a
dxdθ=a(1cosθ),dydθ=asinθ
dydxx=π/2=dydθ|π/2dxdθ|π/2=a(1cosπ/2)asinπ/2=1
Equation of the tangent at θ=π2 is ya=1(xa2(π2))
2x2ya(π4)=0
As we know that
Distance point (x1,y1) from line ax+by+c=0 is |ax1+by1+c|a2+b2
Length of perpendicular drawn from the origin will be the distance of the line from origin
Length of the perpendicular is a(4π)22+(2)2=2a4(π4)

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