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.)( −9a , 3π2​ ) (smaller r value)( 9a , π2​ ) (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 of r with respect to 𝜃 is zero.

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

Setting this equal to zero gives cos(𝜃) = 0, which has solutions 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).

(b) The points of vertical tangency to the polar curve occur when the derivative of r with respect to 𝜃 is undefined.

The derivative dr/d𝜃 = 9a cos(𝜃) is undefined when cos(𝜃) is not defined, which never happens for real values of 𝜃.

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