6.3 Ellipses 37514 Figure 6.393 illustrates the fact that the line Y1P1 from a vertex Y1 to
a point P1 on anellipse is parallel to the line OE from the center of the ellipse to
the point E where the tangent at P1 intersects the tangent at the other vertex Y2.
xl
Show that the coordinates of E are(a,
b912
(1-a)) and prove the fact.Figure 6.393 Figure 6.39415 Figure 6.394 shows a part of the ellipse having, as usual, the standard
equation
(1)
in which 0 < b < a. Let G be the point in the first quadrant where the ellipse
is intersected by the line through the focusFparallel to the directrix. Prove that,
as the figure shows, the coordinates of G are (ae, b2/a). Remark: The result can
he obtained by use of the fact that the y coordinate of G is the product of a and
the distance (ale - ae) from G to the directrix. It can also be obtained by put-
ting x = ae in (1). In each case, it is necessary to use the relation b'- = a2(1 - e2).
16 Using the notation and results of Problem 15, show that the equation of
the tangent to the ellipse at G is ex + y = a. Remark: This shows that the
tangent intersects the y axis at the point (a,0) and intersects the x axis where
the directrix does. These results are illustrated in Figure 6.394. The circle
which has its center at the center of the ellipse and contains the vertices is called
the major circle of the ellipse. Thus the tangent at G intersects the major axis
where the directrix does and intersects the minor axis where the major circle does.
This is one of many elegant geometric theorems that have fascinated men for
centuries.
17 Supposing that P(x,y) is a point on the ellipse having the standard equa-
tion (6.36) which we should now know, use the information in Figure 6.38 (which
we should remember) and the distance formula to show that
lFIPl = a2+ a2-b2x'
a aa2 -1/a2_-_b 2 x
FzPl =Then seek a way in which formulas involving eccentricity can be used to obtain
the same result. Note that IF1PI + IF2PI is what it should be.