simultaneously, to obtain t 0 = tan −^1
(
ay 0
bx 0
)
. The derivative vector
dr
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)