gropu x with x and y with y and complete the squares
(4x^2-48x)+(y^2-4y)+48=0 4(x^2-12x)+(y^2-4y)+48=0 take 1/2 of b and squaer it and ad negative and positive inside 4(x^2-12x+36-36)+(y^2-4y+4-4)+48=0 factor perfect sqare 4((x-6)^2-36)+((y-2)^2-4)+48=0 distribut 4(x-6)^2-144+(y-2)^2-4+48=0 4(x-6)^2+(y-2)^2-100=0 add 100 both sides 4(x-6)^2+(y-2)^2=100 divide both sides by 100 [tex] \frac{(x-6)^2}{25} + \frac{(y-2)^2}{100} =1[/tex] [tex] \frac{(x-6)^2}{5^2} + \frac{(y-2)^2}{10^2} =1[/tex] it is a vertical ellipse with center at (6,2), focci at (6,2+5√3) and (6,2-5√3) major axises at (6,12) and (6,-8) minor axises at (1,2) and at (11,2)
