Hello, chaddy!
Find an equation of the set of points in the Cartesian plane
that are equidistant from the point $\displaystyle (1,1)$ and the line $\displaystyle y=x.$ Code:

 P
 *(x,y) *
/ \ *
A /  \ *
*  \ *
(1,1)  *
 * B
 *
    *        
* 
* 
The distance from point $\displaystyle P(x,y)$ to point $\displaystyle A(1,1)$
. . is given by: .$\displaystyle PA \;=\;\sqrt{(x+1)^2 + (y1)^2} $
The distance from a point $\displaystyle P(x_1,y_1)$ to a line $\displaystyle ax + by + c\:=\:0$
. . is given by: .$\displaystyle d \;=\;\frac{ax_1 + by_1 + c}{\sqrt{a^2+b^2}} $
Hence, the distance from $\displaystyle P(x,y)$ to the line $\displaystyle x  y \:=\:0$
. . is given by: .$\displaystyle PA \;=\;\frac{xy}{\sqrt{1^2 + (1)^2}} \;=\;\frac{xy}{\sqrt{2}}$
Since $\displaystyle PA = PB$, we have: .$\displaystyle \sqrt{(x+1)^2 + (y1)^2} \;=\;\frac{xy}{\sqrt{2}} $
Now simplify . . .