I used the exact same command on
both my TI-89 and my TI-Nspire CX CAS and,
in both cases, got two solutions corresponding
to the points on that ellipse
closest to and farthest from (3,3).
My 89 happened to list the closest point first and
my Nspire happened to list the farthest point first.
Closest point: x=1.69, y=2.67 (and lambda=-3.10)
Farthest point: x=-0.331, y=-4.93 (and lambda=40.2)
Calculator command (using z for lambda):
solve(4(x-3)=z*x and 25(y-3)=z*y and 25x^2+4y^2=100,{x,y,z})
Calculator output (which was displayed in one long line):
x=1.69111... and y=2.66942... and z=-3.09591...
or x=-0.331397... y=-4.93088... and z=40.2102...
Note the "or".
On the 89, I had one strange problem:
After I typed in the solve(...) incantation
and then pressed Diamond, then Enter
to ask for numerical solutions, it returned "false,"
perhaps implying it failed to find a solution.
But when I used just Enter, without the Diamond,
thus asking for exact solutions, it returned
decimal approximations for both solutions.