Technical Graphics  Plane Figures  The Ellipse
To Construct a Tangent to an Ellipse at an Angle
This is a method for
constructing a tangent to an ellipse at an angle to the Major
Axis. For this method you need to know where the Focal
Points are. You also need the Major Auxiliary Circle.
Step
1 : Find the Focal Points, (F1 and F2), and draw in the Major Auxiliary Circle, (red circle). 

Step
2 : Now draw a line at the required angle anywhere outside
of the Major Auxiliary Circle. Draw in lines
perpendicular to this line from the Focal Points as shown.
(blue lines) 

Step
3 : Join where these perpendicular lines cross the Major
Auxiliary Circle to give the tangent to the ellipse. 

Step
4 : Finally to find the Point of Contact between the tangent and the ellipse, put the point of your compass at one of the points where a perpendicular line crosses the Major Auxiliary Circle. Use the distance to the nearest Focal Point and draw a semicircle to cut the perpendicular line. From this point draw a line to the opposite Focal Point, and where this new line cuts the ellipse is the Point of Contact , (purple lines) 

And now you are finished. This method can be used for a
tangent at any angle. 

