Question 1208761
<font color=black size=3>
sqrt is shorthand for "square root"
example: sqrt(5) = {{{sqrt(5)}}}


x-y = k solves to y = x-k


Plug this into the other equation to see what happens.
x^2 + y^2 - 9 = 0
x^2 + (<font color=blue>y</font>)^2 - 9 = 0
x^2 + (<font color=blue>x-k</font>)^2 - 9 = 0
x^2 + x^2 - 2kx + k^2 - 9 = 0
2x^2 - 2kx + k^2 - 9 = 0


In terms of the variable x we have this quadratic template.
ax^2 + bx + c = 0
where in this case,
a = 2
b = -2k
c = k^2 - 9


If the original system has one solution for a fixed specific value of k, then the equation  2x^2 - 2kx + k^2 - 9 = 0 must have one solution. 
A quadratic having one solution would only happen when the discriminant is equal to 0.
d = b^2 - 4ac = discriminant


b^2 - 4ac = 0
(-2k)^2 - 4(2)(k^2-9) = 0 .......... plug in the a,b,c values mentioned earlier
4k^2 - 8(k^2-9) = 0
4k^2 - 8k^2 + 72 = 0
-4k^2 + 72 = 0
4k^2 - 72 = 0
4k^2 = 72
k^2 = 72/4
k^2 = 18
k = plus minus sqrt(18)
k = sqrt(18) or k = -sqrt(18)
k = sqrt(9*2) or k = -sqrt(9*2)
k = sqrt(9)*sqrt(2) or k = -sqrt(9)*sqrt(2)
<font color=red>k = 3*sqrt(2) or k = -3*sqrt(2)</font>



Below is a graph of when k = 3*sqrt(2)
This makes the line x-y = k tangent to the circle toward the bottom right portion of the circle.
{{{
drawing(400,400,-5,5,-5,5,
graph(400,400,-5,5,-5,5,-100,-100,y = x-3*sqrt(2)),
circle(0,0,3)
)
}}}
The circle is centered at (0,0) and has radius 3.
<a href="https://www.geogebra.org/calculator">GeoGebra</a> and <a href="https://www.desmos.com/calculator">Desmos</a> are two of many graphing tools I recommend to use.


And here is the graph when k = -3*sqrt(2)
{{{
drawing(400,400,-5,5,-5,5,
graph(400,400,-5,5,-5,5,-100,-100,y = x+3*sqrt(2)),
circle(0,0,3)
)
}}}
We have another situation where the line is tangent to the circle. 
That tangent point is the solution to the system.


If  -3*sqrt(2) < k < 3*sqrt(2), then the line x-y = k intersects the circle at two different locations. 


If k < -3*sqrt(2) or k > 3*sqrt(2), then the line doesn't intersect the circle at all and there are no solutions. 


Here is an interactive Desmos graph to try out
<a href="https://www.desmos.com/calculator/zijda3qbgr">https://www.desmos.com/calculator/zijda3qbgr</a>
Move the slider for parameter k so you can see how the blue line moves around. 


3*sqrt(2) = 4.24264 approximately
</font>