about to only mathematics

Explanation

Since the x-and the y-axes are two perpendicular tangents to the parabola and both meet at the origin, the directrix passes through the origin

Let = mx be the directrix and (h,k) be the focus. Then, FA=AM
`or" "sqrt((h-1)^(2)+k^(2))=|(m)/(sqrt(1+m^(2)))|` (1)
and FB=BN
`or" "sqrt(h^(2)+(k-1)^(2))=|(m)/(sqrt(1+m^(2)))|` (2)
Squaring and adding (1) and (2), we get
`(h-1)^(2)+h^(2)+k^(2)+(k-1)^(2)=1`
`or" "2x^(2)-2x+2y^(2)-2y+1=0`,
which is the required locus.