(^370) Cones and conics
focus and L2 is another directrix of the ellipse E. Thus E has two foci
and two directrices.
We can look at Figure 6.31 and make some informal observations that
we shall not (and perhaps cannot) make precise. If Y, and Y2 are nearly
equidistant from Y, then the ellipse is nearly circular, the foci are close
together and nearly midway between Y, and Y2, and the eccentricity e is,
as (6.231) shows, nearly zero. If we keep Y, where it is and replace Y2 by
a point many miles up on the cone, then the eccentricity will be near 1,
the ellipse will be relatively flat, and the part of the ellipse within a few
miles of Y, would look so much like a part of a parabola that very careful
inspection of this part would be required to enable us to tell whether the
conic is an ellipse or a parabola or a hyperbola.
Our next step is to use Figure 6.31 to obtain the famous string property
(6.32) I F,PI + IF2PI = I Y1Y2I
of the ellipse E, which shows that the sum of the distances from the foci
of an ellipse to a point P on the ellipse has the same constant value for all
points P on the ellipse. Let P be Y, or Y2 or any other point on the
ellipse. The line VP lies on the cone and is tangent to the lower and upper
spheres at points 111 and 12. Then, as was pointed out in Section 6.2,
IPF,I = IPY1I because the two vectors have their tails at the same point
and are tangent to a sphere at their tips. Also, IFF2I = IP42I for the
same reason, the upper sphere now being involved. Therefore,
(6.33) IF,PI + IF2PI = I111PI + IP22I = I1?,Y2I.
Wherever the point P may be on the cone, the number Iis the con-
stant slant height of the segment of the cone that lies between the parallel
planes 9r, and 72i in fact if d is the distance between 7r, andir2, then
I1r,1r2I = d/cos a, where a is the angle at the vertex of the cone. The
points Y,, F,, F2, and Y2 all lie on the line in which 7r intersects the plane
of the paper. The results of setting P = Y,, and then P = Y2, in (6.33)
give
(6.331) IF1YiI + IF2Y1I =
d
cos a
IF7Y2I + IF;Y2I =
d
cos a
and with the aid of Figure 6.31 we can put this in the form
(6.332) 2IT,FjI + IF,-F21 = IF,F2I + 21T2 2I =
This gives the remarkable fact that
(6.333) IY,F1I = IF2Y21
d
cos a
lu
(lu)
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