The standard form of the equation of an ellipse is
#(x-h)^2/b^2+(y-k)^2/a^2=1#
where #(h,k)# is the center of the ellipse#color(green)( rarr(1,-5)#
#a=7# #color(green)(rarr(given)#
The distance between the two foci is #2ae#
#sqrt((x_1-x_2)^2+(y_1-y_2)^2)=2ae#
#sqrt((1-1)^2+(-4-(-6))^2)=2ae#
#2=2xx7xxe#
#e=1/7#
#b^2=a^2(1-e^2)#
#b^2=48#
Now Substitute in the equation's standard form
#(x-1)^2/48+(y-(-5))^2/49=1#
#(x-1)^2/48+(y+5)^2/49=1#
#color(blue)("We use this form as the ellipse is vertical while we would use this form "#
#color(blue)("if it were horizontal " (x-h)^2/a^2+(y-k)^2/b^2=1#
#color(green)("And to determine whether it's vertical or horizontal you could use the graph"#
graph{(x-1)^2/48+(y+5)^2/49=1 [-14.72, 17.32, -13, 3.02]}
#color(green)("Or you could identify it using the given points as in this question the foci are (1,-6) and (1,-4) so the difference between them is in the " y " value so it's vertical"#
