In the plane rectangular coordinate system xoy, it is known that P is the moving point on the image of the function f (x) = ex (x > 0). The tangent l of the image at point P intersects the y-axis at point m, and the vertical line passing through point P intersects the y-axis at point n. suppose the ordinate of the midpoint of line Mn is t, then the maximum value of T is______ ．

In the plane rectangular coordinate system xoy, it is known that P is the moving point on the image of the function f (x) = ex (x > 0). The tangent l of the image at point P intersects the y-axis at point m, and the vertical line passing through point P intersects the y-axis at point n. suppose the ordinate of the midpoint of line Mn is t, then the maximum value of T is______ ．

Let the coordinate of tangent point be (m, EM) ｜ the equation of tangent line L of the image at point p be y-em = EM (x-m) let x = 0, and the tangent equation of the vertical line of y = (1-m) em passing through point P as l is y-em = - E-M (x-m) let x = 0, and the ordinate of the midpoint of line Mn is t = 12 [(2-m) em + me-m] t '= 12 [-...]

If the distance from the moving point m to the straight line x-2y-3 = 0 is equal to 1 / 2 of the distance from the moving point m to the point (1,2), then the trajectory of the point m is_______

The ratio of the distance to (1,2) divided by the distance to x-2y-3 = 0 = 2
That is, the ratio of the distance to a fixed point divided by the distance to a fixed line is 2
This ratio is the eccentricity e > 1
So it's a hyperbola

In the plane, given the fixed points a, B and the absolute value of AB = 6, if the ratio of the distance from the moving point P to the point a and the distance to the point B is 2:1, the trajectory equation of the moving point P is obtained

Let o in AB be the origin, a and B be (- 3,0), (3,0) respectively, and p be (x, y)
√[（x+3)^2+y^2]/√[(x-3)^2+y^2]=2,(x-5)^2+y^2=16

On the plane, given the fixed points a, B and ab = 6, if the ratio of the distance from the moving point P to the point a and the distance to the point B is 2:1, the trajectory equation of the moving point P is obtained

Take the coordinates of a and B as a (- 3,0) and B (3,0) respectively
P(x,y)
|OA| :|OB| = 2:1
|OA|^2 :|OB|^2 = 4:1
|OA|^2 = 4|OB|^2
|OA|^2 = (x+3)^2 + y^2
|OB|^2 = (x-3)^2 + y^2
(x+3)^2 + y^2 = 4[(x-3)^2 + y^2 ]
(x-5)^2 + y^2 = 16