When the moving point a moves on the circle x2 + y2 = 1, the trajectory equation of the midpoint of its line with the fixed point B (3,0) is () A. （x+3）2+y2=4B. （x-3）2+y2=1C. （2x-3）2+4y2=1D. （x+3）2+y2=12

When the moving point a moves on the circle x2 + y2 = 1, the trajectory equation of the midpoint of its line with the fixed point B (3,0) is () A. （x+3）2+y2=4B. （x-3）2+y2=1C. （2x-3）2+4y2=1D. （x+3）2+y2=12

Let the midpoint m (x, y), then the moving points a (2x-3, 2Y), ∵ a are on the circle x2 + y2 = 1, ∵ (2x-3) 2 + (2Y) 2 = 1, that is, (2x-3) 2 + 4y2 = 1

Given that point a (4,0) and point B is a moving point on the curve X ^ 2 + y ^ 2-2y = 0, the trajectory equation of the midpoint m of line AB is obtained

Let the coordinates of point m be (x, y). Since m is the midpoint of AB, the coordinates of B are
(2x-4,2y) and point B is on the known curve, the coordinate can be brought into the curve equation

Given the hyperbola X-Y | 2 = 1, the straight line passing through point a (2,1) intersects with the known hyperbola at two points P1 and P2, and the trajectory equation of the midpoint P of line p1p2 is obtained

Let the linear equation over a (2,1) be Y-1 = K (X-2), that is, y = kx-2k + 1
The simultaneous hyperbola x ^ - y ^ / 2 = 1 and the analytical formula of this line, by eliminating y, we can obtain the quadratic equation of one variable about X
(k^-2)x^ - (4k^-2k)x +(4k^-4k+3)=0
And △ = (4K ^ - 2K) ^ - 4 (k ^ - 2) * (4K ^ - 4K + 3) = 24 (k - 2 / 3) ^ + 40 / 3 > 0
Let P1 (x1, Y1), q (X2, Y2) be the intersection points of straight line and hyperbola
Then the two different real roots of the above equation must be the abscissa x1, X2 of the two different intersections P1, P2 of the straight line and the hyperbola
x1+x2=(4k^-2k)/(k^-2) ①
The ordinates of P and Q are expressed by their abscissa respectively
y1=kx1-2k+1
y2=kx2-2k+1
∴y1+y2=k(x1+x2)-4k+2
By substituting formula (1), we get the following result:
y1+y2=(8k-4)/(k^-2) ②
According to the formula of midpoint coordinates, the coordinates of M (x, y) in p1p2 can be obtained as follows:
x=(x1+x2)/2
y=(y1+y2)/2
The results are as follows
x=(2k^-k)/(k^-2)
y=(4k-2)/(k^-2) ③
By comparing the two formulas, it is concluded that:
x/y=k/2
k=2x/y
By substituting this formula into (3), we get the following results
(x-1)^/(7/8) - (y-1/2)/(7/4) =1
(in the process of simplification, y is deleted from both sides of the equation, because it can be seen from the image that y cannot always be zero.)
That is, the trajectory of P is a hyperbola with the center at (1,1 / 2) and the intersection on the x-axis