Begin2.DVI

simultaneously, to obtain t 0 = tan −^1

(

ay 0

bx 0

)

. The derivative vector

d
r

dt

=−asin tˆe 1 +bcos tˆe 2

evaluated at the value t 0 , represents a tangent vector to the ellipse at the point P.

A unit vector in the direction of the tangent line at the point P is then given by

ˆet=−asin t

ˆe 1 +bcos tˆe 2

√

a^2 sin^2 t+b^2 cos^2 t

where everything is understood to be evaluated at t=t 0. Using vector addition one

can show the vectors
r 1 and
r 2 must satisfy

r (t) +
r 1 =cˆe 1 and
r (t) +
r 2 +cˆe 1 =
0

These equations allow one to express the vectors
r 1 and
r 2 in the form

r 1 =(c−acos t)ˆe 1 −bsin tˆe 2


r 2 =(−c−acos t)eˆ 2 −bsin tˆe 2

where again, these vectors are to be evaluated at the parameter value t 0.

Unit vectors in the directions of
r 1 and
r 2 can be expressed

ˆer 1 =(√c−acos t)ˆe^1 −bsin tˆe^2

(c−acos t)^2 +b^2 sin^2 t

ˆer 2 =−√(c+acos t)ˆe^1 −bsin tˆe^2

(c+acos t)^2 +b^2 sin^2 t

By employing the definition of the dot product of unit vectors one can verify that

ˆer 1 ·(−ˆet) = cos β=asin t(c−acos t) + b

(^2) sin tcos t
r 1
√
a^2 sin^2 t+b^2 cos^2 t
ˆer 2 ·(ˆet) = cos α=asin t(c+acos t)−b
(^2) sin tcos t
(2 a−r 1 )
√
a^2 sin^2 t+b^2 cos^2 t

where r 1 =|r 1 |=

√

(c−acos t)^2 +b^2 sin^2 t. If cos α= cos βfor all values of the parameter

t, then one must show that

asin t(c−acos t) + b^2 sin tcos t

r 1

√

a^2 sin^2 t+b^2 cos^2 t

=

asin t(c+acos t)−b^2 sin tcos t

(2 a−r 1 )

√

a^2 sin^2 t+b^2 cos^2 t

(7 .7)