Let the point on the locus be #(x,y)#, sum of whose distances from #(3,1)# and #(-1,1)# is #6#, hence

#sqrt((x-3)^2+(y-1)^2)+sqrt((x+1)^2+(y-1)^2)=6# or

#sqrt((x-3)^2+(y-1)^2)=6-sqrt((x+1)^2+(y-1)^2)#

Squaring each side, we get

#(x-3)^2+(y-1)^2=36+(x+1)^2+(y-1)^2-12sqrt((x+1)^2+(y-1)^2)# or

#x^2-6x+9+y^2-2y+1=36+x^2+2x+1+y^2-2y+1-12sqrt((x+1)^2+(y-1)^2)# or

#-6x+9=36+2x+1-12sqrt((x+1)^2+(y-1)^2)# or

#12sqrt((x+1)^2+(y-1)^2)=36+2x+1+6x-9=8x+28# or

#3sqrt((x+1)^2+(y-1)^2)=2x+7# and squaring again

#9((x+1)^2+(y-1)^2)=4x^2+28x+49# or

#9(x^2+2x+1+y^2-2y+1)=4x^2+28x+49# or

#5x^2+9y^2-10x-18y-31=0#

As the coefficient of #x^2# and #y^2# are positive but different, it is an ellipse.

graph{5x^2+9y^2-10x-18y-31=0 [-10, 6, -5, 5]}