Find the points of vertical tangency to the polar curve.

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)

Find the points of horizontal tangency to the polar curve.r = a sin(𝜃)      0 ≤ 𝜃 < 𝜋, a > 0

Find the points of horizontal tangency to the polar curve.r = 2 csc(𝜃) + 3      0 ≤ 𝜃 < 2𝜋

Find dydx and the slopes of the tangent lines shown on the graph of the polar equation. (If an answer does not exist, enter DNE.)r = 2(1 − sin(𝜃))The polar coordinate system is overlaid on the rectangular coordinate system. A curve and three lines are on the graph.Stated points for the curve are given as (x, y) in the rectangular coordinate plane, the cardioid has a dimple at the top of the graph at (0, 0), then proceeds clockwise through the points (2, 0), (0, −4), (−2, 0), and back to the starting point.The first line is tangent to the curve at the point with polar coordinates (2, 0) which is located on the positive horizontal axis, the line begins in the first quadrant, goes down and right, touches the curve and ends in the fourth quadrant.The second line is tangent to the curve at the point with polar coordinates (4, 3𝜋⁄2) which is located on the negative vertical axis, the line begins in the third quadrant, goes horizontally right, touches the curve and ends in the fourth quadrant.The third line is tangent to the curve at the point with polar coordinates (3, 7𝜋⁄6) which is located in the third quadrant, the line begins in the second quadrant, goes vertically down, touches the curve and ends in the third quadrant.dydx= at (2, 0) dydx= at 3, 7𝜋6dydx= at 4, 3𝜋2

Sketch a graph of the polar equation and find the tangent lines at the pole for 0 ≤ 𝜃 < 𝜋 (if any). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)r = 3 cos(3𝜃)