Very nice handwriting but the math and English are confusing.
Let's assume we're told
[tex]\displaystyle 45 = \sum_{i=1}^9 (x_i - 10)^2 [/tex]
The subscript is important.
I think we're told the similar sum with 11 gives the smallest possible value for the sum. This is a rather cagey way of telling us 11 is the mean of the nine points. The mean is the number which minimizes the sum of squared deviations.
[tex]\displaystyle 45 = \sum_{i=1}^9 (x_i - 10)^2 = \sum x_i^2 - 20 \sum x_i + 9(100)[/tex]
[tex]\displaystyle \sum x_i^2= 20 \sum x_i - 900[/tex]
If 11 is the mean, the sum of the points is 9(11)=99.
[tex]\displaystyle \sum x_i^2= 20 (99) - 900 = 1080[/tex]
Answer: 1080
