Hello, nerdzor!
Find the equation of the locus of the point P(x,y) which moves so that
it is always equidistant from the point and the line First, make a sketch . . . Code:

 *S(1,2)
 \
 \
+\
 *P(x,y)
 :
 :
 :
4+    *    L
 Q
The distance from is: .
The distance from to line is: .
Since , we have: .
That is the equation of the locus, so technically, we're done.
. . But, of course, we are expected to simplify our equation.
. . Square both sides: .
. . Expand: .
. . And we have: .
. . Add 1 to both sides: .
. . Factor: .
The locus is a parabola.
. . It opens upward, and its vertex is at