Consider the polar curve defined below.r = 9a sin(𝜃)(a) Find the points of horizontal tangency to the polar curve. (Use pi for 𝜋 as necessary.)( , ) (smaller r value)( , ) (larger r value)(b) Find the points of vertical tangency to the polar curve.( , ) (smaller 𝜃 value)( , ) (larger 𝜃 value)

(a) The points of horizontal tangency to the polar curve occur when the derivative dr/dθ = 0.

The derivative of r = 9a sin(θ) with respect to θ is dr/dθ = 9a cos(θ).

Setting this equal to zero gives 9a cos(θ) = 0, which implies cos(θ) = 0.

The solutions to this equation are θ = π/2 and θ = 3π/2.

Substituting these values back into the equation for r gives r = 9a sin(π/2) = 9a and r = 9a sin(3π/2) = -9a.

So the points of horizontal tangency are (9a, π/2) and (-9a, 3π/2), with -9a < 9a.

(b) The points of vertical tangency to the polar curve occur when the derivative dr/dθ is undefined.

The derivative dr/dθ = 9a cos(θ) is undefined when cos(θ) is undefined, which never happens.

Therefore, there are no points of vertical tangency to the polar curve.