[2*a*t/(1+t^2),b*(1-t^2)/(1+t^2)]

x*(2*a*t/(1+t^2))/a^2+y*(b*(1-t^2)/(1+t^2))/b^2=1

;Simp(#2)
2*t*x/(a*(t^2+1))+y*(1-t^2)/(b*(t^2+1))=1

SOLVE(2*t*x/(a*(t^2+1))+y*(1-t^2)/(b*(t^2+1))=1,y)

;Simp(#4)
[y=b*(2*t*x-a*(t^2+1))/(a*(t^2-1))]

x^2+y^2=1

;Sub(#6)
x^2+(b*(2*t*x-a*(t^2+1))/(a*(t^2-1)))^2=1

EXPAND(x^2+(b*(2*t*x-a*(t^2+1))/(a*(t^2-1)))^2=1,Rational,x)

;Simp(#8)
x^2*(a^2*(t^2-1)^2+4*b^2*t^2)/(a^2*(t^2-1)^2)-4*b^2*t*x*(t^2+1)/(a*(t^2-1)^2)~
+b^2*(t^2+1)^2/(t^2-1)^2=1

SOLVE(x^2*(a^2*(t^2-1)^2+4*b^2*t^2)/(a^2*(t^2-1)^2)-4*b^2*t*x*(t^2+1)/(a*(t^2~
-1)^2)+b^2*(t^2+1)^2/(t^2-1)^2=1,x)

;Simp(#10)
[x=a*((t+1)*(t-1)*SQRT(4*b^2*t^2-a^2*(b^2*(t^4+2*t^2+1)-t^4+2*t^2-1))+2*b^2*t~
*(t^2+1))/(a^2*(t+1)^2*(t-1)^2+4*b^2*t^2),x=a*(2*b^2*t*(t^2+1)-(t+1)*(t-1)*SQ~
RT(4*b^2*t^2-a^2*(b^2*(t^4+2*t^2+1)-t^4+2*t^2-1)))/(a^2*(t+1)^2*(t-1)^2+4*b^2~
*t^2)]

[y=b*(2*t*x-a*(t^2+1))/(a*(t^2-1))]

;Simp(User)
[y=b*(2*t*SQRT(4*b^2*t^2-a^2*(b^2*(t^4+2*t^2+1)-t^4+2*t^2-1))+a^2*(t+1)*(1-t)~
*(t^2+1))/(a^2*(t+1)^2*(t-1)^2+4*b^2*t^2)]