I was attempting to solve one question of maximum solution when something delusive occurred.
It shall be the original question below.
It is known ellipse C:$x^2/a^2+y^2/b^2=1$(a>b>0) passes through point P(2/3,2).
The left and right focus points are $F1$ and $F2$ respectively is the original point and $|PF1|+|PF2|=4.$
then,
It is known that the right vertex A of ellipse C intersects the straight line $l$ at points $M$ and $N$, and the circle with MN as the diameter passes through point A. Find $max(|AM|*|AN|)$.
and I decided to express coordinates by length.
let AM=$t_1$,AN=$t_2$
define $∠β$ to be the intersection angle between $AM$ and the x-axis.
Additionally, I have got AM⊥AN from the condition.
hence M(2-$t_1$,$t_1$sinβ),N(2-$t_1$,$t_2$cosβ).
which entails,
sinβ/$t_2$+cosβ/$t_1$=4/5.
but I am being stuck to get the maximum of t1*t2 by it.
Could anyone help me get one solution?