Does the equation x^2-6x+y^2-6y = - 9 intersects the y-axis

Intersecting the y axis means x is o, so you plug x=0 in the equation and solve for y. if y has real solutions, then the answer is yes.

Let's find out:

0+y^2-6y=-9

y^2-6y+9=0

(y-3) (y-3) = 0

y=3

3 is a real number, so yes, the equation intersects the y axis at (0,3)