Answer:
b. ellipse; [tex]3x^2+y^2+6x-8y+13=0[/tex]
Step-by-step explanation:
The given equation is
[tex]4y^2+12x^2=24[/tex]
Divide through by 24;
[tex]\frac{y^2}{6}+\frac{x^2}{2}=1[/tex]
This is an ellipse that has its center at the origin.
The ellipse is translated from the origin to (-1,4).
The equation of the translated ellipse is
[tex]\frac{(y-4)^2}{6}+\frac{(x+1)^2}{2}=1[/tex]
Multiply through by 6.
[tex](y-4)^2+3(x+1)^2=6[/tex]
Expand;
[tex]y^2-8y+16+3(x^2+2x+1)=6[/tex]
[tex]y^2-8y+16+3x^2+6x+3=6[/tex]
This implies that;
[tex]3x^2+y^2+6x-8y+13=0[/tex]
