Answer:
[tex]\dfrac{(x-1)^2}{4}+\dfrac{y^2}{25}=1[/tex]
Step-by-step explanation:
The equation of the ellipse is
[tex]\dfrac{(x-x_0)^2}{a^2}+\dfrac{(y-y_0)^2}{b^2}=1,[/tex]
where [tex](x_0,y_0)[/tex] are the coordinates of the center.
If the vertices of an ellipse are at A(1, 5) and B(1, -5), then the center is the midpoint of the segment AB. Hence, the center has coordinates
[tex]\left(\dfrac{1+1}{2},\dfrac{5+(-5)}{2}\right)=(1,0).[/tex]
The coordinates of the vertices satisfy the equation:
[tex]\dfrac{(1-1)^2}{a^2}+\dfrac{(5-0)^2}{b^2}=1\Rightarrow b^2=25.[/tex]
If (3, 0) is a point on the ellipse, then its coordinates satisfy the equation:
[tex]\dfrac{(3-1)^2}{a^2}+\dfrac{(0-0)^2}{b^2}=1\Rightarrow a^2=4.[/tex]
Therefore, the equation of the ellipse is
[tex]\dfrac{(x-1)^2}{4}+\dfrac{y^2}{25}=1.[/tex]
