Tangents and normals Let a polar curve be described by r = f(θ) and let ℓ be the line tangent to the curve at the point P(x, y) = P(r, θ) (see figure).

a. Explain why tan α = dy/dx.

b. Explain why tan θ = y/x.

c. Let φ be the angle between ℓ and OP. Prove that tan φ = f(θ)/f′(θ).

d. Prove that the values of θ for which ℓ is parallel to the x-axis satisfy tan θ = −f(θ)/f′(θ).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f′(θ)/f(θ).

Solution 50AE

a. Here α is an angle made by the line with positive x -axis

Therefore tan α is the slope of the tangent line

But we know that slope of the tangent to the curve r = f (is

tan=

b. In the above figure distance from the point to x - axis is y units

And the distance from the point to x-axis is x units

From the right angle triangle

tan=

c. ...