If x² + y² − 10x + 26y + 194 = 0, then the value of x² + y² is:

Correct option is B

​Given:

​$$x^2 + y^2 - 10x + 26y + 194 = 0$$​​

We are to find the value of $$x^2 + y^2$$​​

Formula Used: 

​$$(a+b)^2 = a^2 +b^2 +2ab \\ \ \\ (a-b)^2 = a^2+b^2-2ab$$​​

Solution: 

​$$x^2 + y^2 - 10x + 26y + 194 = 0 \\ \ \\x^2 - 10x + y^2 + 26y + 194 = 0 \\ \ \\x^2 - 10x + 25 + y^2 + 26y + 169 = 0 \\ \ \\ x^2 - 10x + (5)^2 + y^2 + 26y + (13)^2 = 0 \\ \ \\(x - 5)^2 + (y + 13)^2 = 0$$​​

Since a sum of squares is zero, both must be zero:

​$$(x - 5)^2 = 0 \Rightarrow x = 5$$​​

​$$(y + 13)^2 = 0 \Rightarrow y = -13$$​​

Now,

​$$x^2 + y^2 = 5^2 + (-13)^2 = 25 + 169 = 194$$​​