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a^2/6 +b^2/3 =1
a^2+2b^2 =6 (1)
(b+1)/(a-4)= -a/(2b)
2b^2+ 2b = -a^2 +4a
(6-a^2) + 2b = -a^2 +4a
2b= 4a -6
b= 2a-3 (2)
sub (2) into (1)
a^2+2b^2 =6
a^2+2(2a-3)^2 =6
9a^2-24a + 18 =6
9a^2-24a + 12 =0
3a^2-8a + 4 =0
(3a-2)(a-2) =0
a=2 or 2/3
from (2)
a=2 , b=2a-3 = 1
a=2/3 , b=2a-3 = 4/3 -3 = -5/3
ie
(a,b)= (2,1) or ( 2/3, -5/3)
a^2+2b^2 =6 (1)
(b+1)/(a-4)= -a/(2b)
2b^2+ 2b = -a^2 +4a
(6-a^2) + 2b = -a^2 +4a
2b= 4a -6
b= 2a-3 (2)
sub (2) into (1)
a^2+2b^2 =6
a^2+2(2a-3)^2 =6
9a^2-24a + 18 =6
9a^2-24a + 12 =0
3a^2-8a + 4 =0
(3a-2)(a-2) =0
a=2 or 2/3
from (2)
a=2 , b=2a-3 = 1
a=2/3 , b=2a-3 = 4/3 -3 = -5/3
ie
(a,b)= (2,1) or ( 2/3, -5/3)
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