we have
49x^2 + 16y^2 - 392x +160y + 400 = 0
Complete the square
Group terms
[tex](49x^2-392x)+(16y^2+160y)=-400[/tex]Factor 49 and 16
[tex]49(x^2-8x)+16(y^2+10y)=-400[/tex][tex]49(x^2-8x+16)+16(y^2+10y+25)=-400+16(49)+25(16)[/tex][tex]\begin{gathered} 49(x^2-8x+16)+16(y^2+10y+25)=784 \\ 49(x^{}-4)^2+16(y+5)^2=784 \end{gathered}[/tex]Divide by 784 both sides
[tex]\begin{gathered} 49(x^{}-4)^2+16(y+5)^2=784 \\ \frac{49(x^{}-4)^2}{784}+\frac{16(y+5)^2}{784}=1 \end{gathered}[/tex]simplify
[tex]\frac{(x^{}-4)^2}{16}+\frac{(y+5)^2}{49}=1[/tex]we have a vertical elipse
the center is the point (4,-5)
major semi axis is 7
we have
a^2=16 --------> a=4
b^2=49 ------> b=7
Find the value of c
[tex]\begin{gathered} c=\sqrt[]{b^2-a^2} \\ c=\sqrt[]{33} \end{gathered}[/tex]see the attached figure to better understand the problem
