Answer:
Option b
Step-by-step explanation:
The equation [tex]4x ^ 2 + 5y ^ 2 = 20[/tex] has center in (0,0).
But the transformation [tex]T(5, -6)[/tex] shifts the center of the equation to the point (5, -6).
Therefore, when applying [tex]T(5, -6)[/tex] we will have the following equation translated.
[tex]4(x-5) ^ 2 + 5(y - (-6)) ^ 2 = 20[/tex].
Simplifying we have:
[tex]4(x-5) ^ 2 + 5(y + 6) ^ 2 = 20[/tex]
Now we expand [tex](x-5) ^ 2[/tex] and [tex](y + 6) ^ 2[/tex]
[tex]4(x ^ 2 -10x +25) + 5(y ^ 2 + 12y +36) = 20\\\\4x ^ 2 -40x + 100 + 5y ^ 2 + 60y + 180 = 20\\\\4x ^ 2 + 5y ^ 2 -40x + 60y +260 = 0[/tex]
The equation of a circle has the form
[tex]h(x-a) ^ 2 + q(y-b) ^ 2 = r[/tex]
For h = 1 and q = 1.
If [tex]h \neq 1[/tex] and [tex]q\neq 1[/tex] then the graph becomes an ellipse.
In this problem h = 4 and q = 5 therefore the figure is an ellipse
