Rewrite the second equation: (x-1)2+(y+4)2/4=25.
From the first equation (x-1)2=25-(y+4)2, so: 25-3(y+4)2/4=25, from which y=-4.
Therefore (x-1)2=25, x-1=5 or x-1=-5, making x=6 or -4.
Solutions are (x,y)=(-4,-4), (6,-4).
Graphically this is a circle within an ellipse. The conics are concentric and the circle touches the ellipse tangentially.