Answer :
Given:
Major axis = 10
Foci at (-1,0) and (-7,0)
Find-: Equation of ellipse.
Sol:
The major axis length is 10.
That mean:
[tex]\begin{gathered} 2a=10 \\ \\ a=\frac{10}{2} \\ \\ a=5 \end{gathered}[/tex]Distance between two foci is:
[tex]\begin{gathered} D=|-7-(-1)| \\ \\ D=|-7+1| \\ \\ D=6 \end{gathered}[/tex]So,
[tex]\begin{gathered} 2ae=6 \\ \\ ae=\frac{6}{2} \\ \\ ae=3 \\ \\ e=\frac{3}{5} \end{gathered}[/tex]The value of "b" is:
[tex]\begin{gathered} b^2=a^2(1-e^2) \\ \\ b^2=25(1-\frac{9}{25}) \\ \\ b^2=25-9 \\ \\ b^2=16 \end{gathered}[/tex]Center of ellipse = Midpoint of foci
[tex]\begin{gathered} =(\frac{-1+(-7)}{2},\frac{0+0}{2}) \\ \\ =(-4,0) \end{gathered}[/tex]Hence, the equation of the ellipse is:
[tex]\frac{(x+4)^2}{25}+\frac{y^2}{16}=1[/tex]