Given that the projection of the point m on the x-axis of the equiaxed hyperbola x2-y2 = R2 is n, then the trajectory equation of the midpoint P of the segment Mn is n______ ．

Let P (x, y) be the midpoint of line Mn, so the coordinate of M is (x, 2Y). Because m is on the equiaxed hyperbola x2-y2 = R2, so X2 - (2Y) 2 = R2, so the trajectory equation of the midpoint P of line Mn is: x2-4y2 = R2. So the answer is: x2-4y2 = R2

Make the secant op1p2 of the curve y = x & # 178; + 1 through the origin, and find the trajectory equation of the midpoint P of the chord p1p2,

Let: P1 (x1, Y1) P2 (X2, Y2) midpoint (x, y)
Then X1 + x2 = 2x, Y1 + y2 = 2Y
Points P1 and P2 are on the curve y = x2 + 1
y1=x1^2+1
y2=x2^2+1
The difference between two formulas: y1-y2 = (x1-x2) (x1 + x2)
(y1-y2)/(x1-x2)=k=x1+x2=2x
And because the line passes through the origin and the midpoint, k = Y / X
Y / x = 2x y = 2x ^ 2 (x is not zero)
When x = 0, the midpoint coincides with P1 and P2, which also satisfies the requirement
So the midpoint trajectory is y = 2x ^ 2

Point B (3,0), point a moves on the curve X ^ + y ^ = 1, then the trajectory equation of point P in line AB is

Use the substitution method
Let P (x, y), a (x1, Y1),
Then x = (x1 + 3) / 2, y = (Y1 + 0) / 2,
The solution is X1 = 2x-3, Y1 = 2Y,
Substituting into the known curve equation, (2x-3) ^ 2 + (2Y) ^ 2 = 1,
It is reduced to (x-3 / 2) ^ 2 + y ^ 2 = 1 / 4,
It is a circle with a center of (3 / 2,0) and a radius of 1 / 2

Given that the projection of the point m on the x-axis of the equiaxed hyperbola x2-y2 = R2 is n, then the trajectory equation of the midpoint P of the segment Mn is n______ ．