The general equation of an ellipse is x^2/a^2+y^2/b^2=1. This ellipse is centred at (0,0).
Plug in (1,√3/2): 1/a^2+3/(4b^2)=1, so 4b^2+3a^2=4a^2b^2 and 4b^2(a^2-1)=3a^2, so b^2=3a^2/(4(a^2-1)).
The second point is ambiguous. As written it is effectively (√2,1) but we won't assume the y coord and we'll just call it Y. Plug in (√2,Y): 2/a^2+Y^2/b^2=1⇒2/a^2+4Y^2(a^2-1)/3a^2=1; 6+4Y^2(a^2-1)=3a^2; 6+4Y^2a^2-4Y^2=3a^2; 6-4Y^2=a^2(3-4Y^2), a^2=2(3-2Y^2)/(3-4Y^2). If Y=1 then a^2 would be negative and we wouldn't have an ellipse. In fact, we know (6-4Y^2)/(3-4Y^2)>0. This means 3-2Y^2>0, Y<√(3/2) and 3-4Y^2>0, Y<√3/2; or 3-2Y^2<0 and 3-4Y^2<0, meaning Y>√(3/2) and Y>√3/2. Therefore, since 3/2>3/4 Y>√(3/2) (1.22 approx) or Y<√3/2 (0.87 approx). Y cannot be between these values and it can equal neither of them. Let's assume Y is √2/2 (0.71 approx). Then a^2=4, making b^2=1.
The equation of the ellipse is x^2/4+y^2=1 or x^2+4y^2=4. Both given points satisfy this equation: 1+4(3/4)=4=2+4/2.