Traditional Culture Encyclopedia - Photography and portraiture - The distance ratio between the coordinates of point M and two fixed points A and B is a positive number M, and the trajectory equation of m is found.

The distance ratio between the coordinates of point M and two fixed points A and B is a positive number M, and the trajectory equation of m is found.

Let A(-t, 0) B(t, 0) M(x, y)

|MA|=√[(x+t)^2+y^2]

|MB|=√[(x-t)^2+y^2]

The ratio of the coordinates of point M to the distance between two fixed points A and B is positive m,

√[(x+t)^2+y^2]/√[(x-t)^2+y^2]=m

(x+t)^2+y^2=m^2[(x-t)^2+y^2)

x^2+2xt+t^2+y^2=m^2x^2-2m^2tx+m^2t^2+y^2t^2

(m^2- 1)x^2-2t(m^2+ 1)x+(m^2- 1)y^2+(m^2- 1)t=0

When m 2-1= 0, that is, m = 1, the trajectory of m is a straight line.

The trajectory of m 2- 1 ≠ 0 m is a circle.