Question: A point P(a,b) is equidistant from the y-axis and from the point (4,0). find a relationship between a and b

My attempt

using the formula to find the coordinates on y-axis

find the a

since the coordinate is on the y-axis therefore x= 0

hence

therefore a = 2

since (4,0) P(a,b)

therefore y=2b

the coordinates on the y-axis is (0,2b) let 's call it b and a= (4,0)

find the distance of BP and AP

find AB

AB = AP + BP

the answer in the book is

where did i go wrong?

please any help