Find the points of horizontal and vertical tangency (if any) to the polar curve.

To find a tangent line to a polar curve, we regard as parameter and write it's parametric equations as,

We are given the polar curve 

Now let's convert polar equation into parametric equation,

Slope of the line tangent to the parametric curve is given by the derivative 

Let's take the derivative of x and y with respect to 

To find a tangent line to a polar curve, we regard as parameter and write it's parametric equations as,

We are given the polar curve 

Now let's convert polar equation into parametric equation,

Slope of the line tangent to the parametric curve is given by the derivative 

Let's take the derivative of x and y with respect to 

use the trigonometric identity:

Use the trigonometric identity:

We locate horizontal tangents by finding the points where  ( provided that )

and vertical tangents at the points where ( provided that )

Setting the derivative of x equal to zero for locating vertical tangents,

Let's find the corresponding radius r for the above angles,

For 

 

For 

For 

For 

Now let's set the derivative of y equal to zero for locating horizontal tangents,

Now, find the corresponding radius r for above angles,

For 

For 

For 

For 

Note: If we plot the polar curve , its a circle and it should have two horizontal and two vertical tangents. However we got four points because it depends on a, whether it's positive or negative.

For positive value of a ,

the polar curve has horizontal tangents at 

and vertical tangents at 

For negative value of a,

the polar curve has horizontal tangents at 

and vertical tangents at