Calculus: Analytic Geometry and Calculus, with Vectors

374 Cones and conics

coordinate axes, find the equation of the line T1 tangent to the ellipse at
P1. Ins.:
XIX

a2

+ UZ = 1.

9 Find the coordinates of the points where the tangent T1 of Problem 8

intersects the coordinate axes. Ins.:

\xl'O)' (0")

10 Find the coordinates of the points where the tangent Ti of Problem 8

intersects the lines through the foci perpendicular to the major axis. llns.:

(-ae,

yl +eal)), \ae' y, \1

11 Find the coordinates of the points where the tangent Ti of Problem 8

intersects the directrices of the ellipse. Ins..:

(( 11

(-e'yl\1+aee'yl\1\ ae

12 Let the line T1 tangent to an ellipse at P, intersect a directrix at Q, and

let F be the focus corresponding to the directrix. With the aid of Problem 11

and the fact that b2 = a2(l - e2), prove that the line FQ, is perpendicular to the

line FP1. Remark: This result has some quite surprising consequences. If the

focus F, directrix D, and one single point Pj(xl,yl) of an ellipse are marked in a

plane, we can give a simple rule for drawing the line T which is tangent to the

Figure 6.392

undrawn ellipse at Pi. In case P1 is on the

line through F perpendicular to D, the

tangent T1 is the line through P1 parallel to

D. Otherwise, the tangent T1 is the line

containing P1 and the point Q, where the

line through F perpendicular to the line FP1

intersects the directrix. This result implies

that if P1 and P2 are points at the end of a

P1(xi,yl

'1 focal chord (a chord containing a focus), then

the tangents at P1 and P2 intersect at the point

Q, on the directrix where the line through the

focus perpendicular to the line P1P2 meets the

directrix. Figure 6.392, in which a part of

the ellipse is drawn, illustrates this elegant

geometric fact.

13 Figure 6.392 illustrates another in-

teresting geometric fact. Then line OP1

and the line through F perpendicular to the tangent Ti at P1 intersect ata point

11 on the directrix. Prove the fact by proving that each line intersects the direc-

trix at the point I (ae, ayex1)i.