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.