Apollonius's Ellipse and Evolute Revisited - Moving the Given Point

If the given point O is at the origin, the hyperbola (3) reduces to the coordinate axes and there are four intersection points as already noted above. If O is on the major axis at a distance from the origin just smaller than the major semi-axis, there are only two intersection points, the endpoints of the major axis of the ellipse. Is there a transition from four intersections to just two?

One may begin to investigate this question by moving the point O(x₀,y₀) along a ray from the origin at a fixed nonzero angle q by letting (x₀,y₀)=(rcos⍬,rsin⍬) and varying the distance r from the origin. For example, in Fig. 2, one can continue moving the point outward from the origin along the line ⍬ = -π/4. One can put the equation of the hyperbola (3) into the standard form XY=±a_h²/2 by translating its center to the origin through the substitution (x,y) = (X+x₁,Y+y₁) and setting the linear terms in X and Y to zero to determine its center

Its vertices lie on its major axis X=±Y, with their distance from the center equalling Ö2 times the absolute value of the coordinates of the intersection with this axis, namely

As r increases, this center moves along the line through the origin on the opposite side of the x-axis from the original ray and whose slope is the ratio of the coefficients of r in the coordinates of the center

and therefore makes a smaller angle with the x-axis than the original ray.

Thus the center of the hyperbola moves outward along the second line and its vertices expand outward from it by distances all proportional to r as r increases until the one branch of the hyperbola not passing through the origin moves out of the ellipse and its two intersections with the ellipse degenerate to one and then none. The other branch through the origin must always intersect the ellipse in exactly two points. Thus the four intersections of the hyperbola locating the feet of the normals passing through O degenerate to three when the hyperbola becomes tangent to the ellipse (see Fig. 5), and then two when it moves outside. In the next section we will show exactly where this transition from four to two normals occurs.