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Slope of the line passing normal through the curve x = p( cosθ + θsinθ) and y = p(sinθ - θcosθ) will be:
1. -cotθ
2. -tanθ
3. cotθ
4. tanθ

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Correct Answer - Option 1 : -cotθ

Formula used:

1. If two line with slope m1 and m2 are perpendicular to each other, then

m1m2 = -1

2. \(\frac{d}{dθ } sinθ\ =\ cosθ\)

3. \(\frac{d}{dθ } cosθ\ =\ -\ sinθ\)

Calculation:

Given that, 

x = p( cosθ + θsinθ)

Differentiating with respect to θ 

dx/dθ = p(-sinθ + sinθ + θcosθ)

⇒ dx/dθ = p(θcosθ)       ----(1)

Also,  y = p(sinθ - θcosθ) 

Differentiating with respect to θ 

dy/dθ = p((cosθ - cosθ + θsinθ)      

⇒ dy/dθ = p(θsinθ)        ----(2)

Divide equation (1) from equation (2)

dy/dx = tanθ

Let, slope of perpendicular line is m

⇒ m(tanθ) = -1      (∵ m1m2 = -1)

⇒ m = - cotθ         (∵ \/tanθ = cotθ)

Hence, option (1) is correct.

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