Now drag F₂ outside the circle. What do you observe, what happens?
Drag or animate point S on the circle and watch P trace the path of the locus …
Note: the extra blue lines are in fact the asymptotes of the hyperbola. It is drawn due to a limitation in most software programs which connects the endpoints of the arms of the hyperbola.

Drag F₂ to different positions outside the circle … Note that |F₁P – F₂P| remains constant.

Now drag F₁, changing the size of the circle – why does the value of |F₁P – F₂P| change?

Can you explain why:

• if F₂ is inside the circle the locus is an ellipse, but
• if F₂ is outside the circle, the locus is an hyperbola.
• What if F₂ is on the circle?

We have already proven that for the ellipse, the construction gives F₁P + F₂P = r = constant.
Now prove that if F₂ is outside the circle, then F₂P – F₁P = r = constant.

Click here to see a proof:   

We define the locus of all points P such that F₂P – F₁P is constant as an hyperbola.

Tangents to the hyperbola
Click “Show angles”, and note that the hyperbola has the same reflective property as the parabola and the ellipse, i.e. the pink line is a tangent to the hyperbola at P.

The hyperbola as an envelope
We can form the hyperbola an envelope of tangents, as we have seen for the parabola and the ellipse.
Click on “Show tangents”. Drag F₁ and F₂ and reflect on what you see!
When is the locus an ellipse, when an hyperbola, when a circle? Beautiful is it not!?