Conics through 4 points and tangent to the y-axis

h[x_,y_]:={x^2,x y,x,y^2,y,1};

pts={{1,2},{2,3},{-1,3},{2,-3}};
conditions=Table[h[p[[1]],p[[2]]],{p,pts}];
ns=NullSpace[conditions];
projline=s*ns[[1]]+t*ns[[2]];
sol=Solve[projline[[5]]^2-4projline[[4]]projline[[6]]==0,{s,t}];
sol={s,t}/.sol;
sol=sol/.{t->1};
coeffs1=projline/.{s->sol[[1,1]],t->1};
coeffs2=projline/.{s->sol[[2,1]],t->1};
conic1=coeffs1.{x^2,x y,x, y^2, y, 1};
conic2=coeffs2.{x^2,x y,x, y^2, y, 1};
c1=ContourPlot[{conic1==0},{x,-5,5},{y,-5,5},ContourStyle->Red];
c2=ContourPlot[{conic2==0},{x,-5,5},{y,-5,5},ContourStyle->Blue];
yaxis=ParametricPlot[{0,y},{y,-5,5},PlotStyle->Thick];
P=Table[{PointSize[Large],Point[p]},{p,pts}];

MatrixForm[conditions]

Show[Graphics[P],Axes->True,PlotRange->5]

Show[c1,c2,yaxis,Graphics[P]]