The circle given by
x^2+y^2-2y-11=0
can be written in standard form like this:
x^2+(y-k)^2=12
.
what is the value of k in this equation?

Typically, the equation of a circle is given like this: (x-h)^2 + (y-k)^2 = r^2, where r is the radius of the circle and the point (h,k) is the center. To find the value k, you need to put the first equation you have into this form. Standard form requires that you only have the1 y variable, as seen in the parentheses, so you first need to complete the square.

x^2 + y^2 - 2y = 11
x^2 + y^2 - 2y +1 = 11 + 1
x^2 + (y - 1)^2 = 12

And that puts the equation in standard form. k = 1.