The ratio of the distance from the moving point P (x, y) to the vertex f (1,0) to the distance from it to the fixed line x = 4 is 1:2,

The ratio of the distance from the moving point P (x, y) to the vertex f (1,0) to the distance from it to the fixed line x = 4 is 1:2,

The distance from the moving point P (x, y) to the vertex f (1,0) is the root of (x-1) ^ 2 + y ^ 2
The distance from P (x, y) to the fixed line x = 4 is | x-4|
The distance ratio is 1:2
The root of (x-1) ^ 2 + y ^ 2: | x-4 | = 1:2
The root of 2 * (x-1) ^ 2 + y ^ 2 = | x-4|
square
4(x-1)^2+4*y^2=(x-4)^2
Just unfold and organize