THE AUXILIARY CIRCLE

WHAT IS THE AUXILIARY CIRCLE?

 

 

The auxiliary circle is the name given to the circle with its centre on the axis, which contains the two vertices. 

Figure 1 shows the auxiliary circle for an ellipse. We have seen the auxiliary circle for the ellipse previously when we used it to construct an ellipse, that time we had a major and minor auxiliary circle but this time we shall only deal with the major.

 

 

 

 

 

 

Figure 2 shows the auxiliary circle for a parabola. 

As discussed earlier (when dealing with tangents) the second vertex for a parabola is at infinity, therefore the radius of the auxiliary circle will be of infinite value and can be represented as a line perpendicular to the axis at the vertex (i.e. A tangent to the curve at the vertex).   

 

 

 

 

 

 

 

 

 

Figure 3 shows the auxiliary circle for a hyperbola.

As with the ellipse the auxiliary circle for a hyperbola is a circle with its centre on the axis and contains the two vertices.

 

 

 

 

 

 

 

COMMON PROPERTY

PRINCIPLE:

A perpendicular to a tangent from a focal point will meet the tangent on the auxiliary circle.

THE ELLIPSE

If a line is drawn from the foci of an ellipse, perpendicular to a tangent it will meet the tangent on the auxiliary circle.

 

 

THE PARABOLA

If a line is drawn from the focus of a parabola, perpendicular to any tangent to the curve, it will meet the tangent on the auxiliary circle. (a tangent at the vertex)

 

THE HYPERBOLA

If a line is drawn from the foci of a hyperbola, perpendicular to any tangent to the curve, it will meet the tangent on the auxiliary circle.