There are various ways to draw an ellipse.
1.
Pin and String: This is the most easy method to draw an ellipse. Place a pin in each focus leaving a string with length 2a.Pull the string tightly with a pencil point and slide the pencil to draw the ellipse.
2.
Trammel method: This is one of the easiest and accurate method to draw an ellipse. Mark a piece of paper with points O, A and B where OB = b, OA = a. Draw the two axes. Slide A along the minor axis and B along the major axis to get points. Point O traces the ellipse.

3.
Foci and axes: When the major and minor axes are given foci can be determined. Foci lies on the major axis at distance a from the endpoint of the minor axis. Draw the arcs with radii a from the endpoints of the minor axes and make a smooth curve with endpoints of both the axes and the points obtained from the arcs.
4.
Drawing an ellipse with two circles: Step 1: Draw a the major axis and divide it into 3 equal parts and name it A, B, C and D.
Step 2: Take the radius as AB and draw 2 circles using B and C as centres. These 2 circles will be crossed each other in 2 points name them as E1 and E2.
Step 3: Draw 2 arcs on each circle by using the compass, with same radius, and by using A and D as centers. Name them as F1, F2, F3 and F4.
Step 4: Take the radius the length between F1 and E2 and run it from F1 to F2 using E2 as center. Repeat this step and draw the line from F3 to F4 by using E1 as center.

Finally make a smooth ellipse.