The general equation of an ellipse is:
[tex]\frac{\mleft(x-h\mright)^2^{}}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex]where a > b.
To get this form, we need to divide by 49 and by 16, the equation provided, as follows:
[tex]\begin{gathered} 49x^2+16y^2=784 \\ \frac{49x^2+16y^2}{49\cdot16}=\frac{784}{49\cdot16} \\ \frac{49x^2}{49\cdot16}+\frac{16y^2}{49\cdot16}=\frac{784}{784} \\ \frac{x^2}{16}+\frac{y^2}{49}=1 \end{gathered}[/tex]This means that the values of the constants are: h = 0, k = 0, b = 4, and a = 7.
Since a is greater than b, then the major axis is the vertical axis. The ends of the major axis are the vertices and they are found as follows:
(h, k+a) = (0, 0+7) = (0, 7)
(h, k-a) = (0, 0-7) = (0, -7)
And the foci are found as follows:
(h, k+c) = (0, 0+5.74) = (0, 5.74)
(h, k-c) = (0, 0-5.74) = (0, -5.74)
where c is computed as follows:
[tex]\begin{gathered} c^2=a^2-b^2 \\ c^2=49-16 \\ c=\sqrt[]{33} \\ c\approx5.74 \end{gathered}[/tex]