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

Parametric curve (x(t),y(t)) has a horizontal tangent if its slope  is zero i.e when and 

It has a vertical tangent if its slope approaches infinity i.e it is undefined, which implies that and  

Given parametric equations are:

Here the parameter is 

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

Use trigonometric identity: 

For Horizontal tangents,...

Parametric curve (x(t),y(t)) has a horizontal tangent if its slope  is zero i.e when and 

It has a vertical tangent if its slope approaches infinity i.e it is undefined, which implies that and  

Given parametric equations are:

Here the parameter is 

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

Use trigonometric identity: 

For Horizontal tangents, set the derivative of y equal to zero

Let's check for the above angles,

For 

For 

So, there are no horizontal tangents.

Now for vertical tangents, set the derivative of x equal to zero,

Let's check for the above angles,

For 

 

For 

For 

For 

So, the curve has vertical tangents at 

Now let's find the corresponding x and y coordinates by plugging in the parametric equation,

For 

For 

So, the given parametric curve has vertical tangent at (4,0).