The general equation of an ellipse is
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]
Where (h, k) are the coordinates of the center.
For our ellipse, we already have the center (2, 1), then, our equation will be
[tex]\frac{(x-2)^2}{a^2}+\frac{(y-1)^2}{b^2}=1[/tex]
Using the points that belong to this ellipse, we can substitute them on this equation to find the coefficients a and b.
First, let's use the point (4, 1). Doing the substitution, we have
[tex]\begin{gathered} \frac{(4-2)^2}{a^2}+\frac{(1-1)^2}{b^2}=1 \\ \frac{(2)^2}{a^2}=1 \\ a=2 \end{gathered}[/tex]
Using the other point, we can calculate the other coefficient.
[tex]\begin{gathered} \frac{(1-2)^2}{4}+\frac{(1+2\sqrt[]{3}-1)^2}{b^2}=1 \\ \frac{(-1)^2}{4}+\frac{(2\sqrt[]{3})^2}{b^2}=1 \\ \frac{1}{4}+\frac{12^{}}{b^2}=1 \\ \frac{12^{}}{b^2}=\frac{3}{4} \\ 4\cdot12=3\cdot b^2 \\ b^2=16 \\ b=4 \end{gathered}[/tex]
And then, we have our ellipse equation.
[tex]\frac{(x-2)^2}{4}+\frac{(y-1)^2}{16}=1[/tex]
