B(0,-b),
M(m,-b),
P(p,q)
P is on the ellipse, p 2/a 2+q 2/b 2 = 1.
( 1)
k(AM)=k(AP)
= & gt
-2b/m=(q-b)/p
(2)
MO⊥PB
= & gt
k(MO)*k(PB)=- 1
= & gt
-b/m*(q+b)/p=- 1
(3)
(2)/(3) Available
2p/(q+b)=-(q-b)/p, available after finishing.
2p^2+q^2=b^2
(4)
(4)-( 1) * b 2, available
2p^2-p^2*b^2/a^2=0
That is 2 = b 2/a 2.
It can be seen that for ellipses, there are
B>a, that is, the long axis is on the y axis.
∴e=c/b, then b 2 = a 2+c 2.
2=b^2/a^2=b^2/(b^2-c^2)= 1/( 1-e^2)
solve
e= 1/√2=√2/2
∴ Eccentricity of ellipse is e=√2/2.