The gravitational force exerted on the moon by the planet when the moon is at maximum distance [tex]r_{max}[/tex] is
[tex]F_{min}=G \frac{Mm}{r_{max}^2} [/tex]
where G is the gravitational constant, M and m are the planet and moon masses, respectively. This is the minimum force, because the planet and the moon are at maximum distance.
Similary, the gravitational force at minimum distance is
[tex]F_{max}=G \frac{Mm}{r_{min}^2}[/tex]
And this is the maximum force, since the distance between planet and moon is minimum.
The problem says that [tex]F_{max}[/tex] exceeds [tex]F_{min}[/tex] by 11%. We can rewrite this as
[tex]F_{max}=(1+0.11)F_{min}=1.11 F_{min}[/tex]
Substituing the formulas of Fmin and Fmax, this equation translates into
[tex] \frac{1}{r_{min}^2}=1.11 \frac{1}{r_{max}^2} [/tex]
and so, the ratio between the maximum and the minimum distance is
[tex] \frac{r_{max}}{r_{min}}= \sqrt{ 1.11 }=1.05 [/tex]
