A math problem (about parabola) Through point a (0, - 1), make a straight line L intersecting parabola y2 = 4x intersecting at two points B and C, and find the trajectory equation of the midpoint P of BC?

A math problem (about parabola) Through point a (0, - 1), make a straight line L intersecting parabola y2 = 4x intersecting at two points B and C, and find the trajectory equation of the midpoint P of BC?

y+1=kx
y=kx-1
Substituting
k^2x^2-(2k+4)x+1=0
x1+x2=(2k+4)/k^2
y=kx-1
So Y1 + y2 = kx1-1 + kx2-1 = K (x1 + x2) - 2 = 4 / K
midpoint
So x = (x1 + x2) / 2, y = (Y1 + Y2) / 2
So y = 2 / K, k = 2 / y
Substituting x = (K + 2) / K ^ 2, K ^ 2x = K + 2
4x/y^2=2/y+2
2x=y^2+y
Because the parabola y ^ 2 = 4x, so x > = 0
Intersection at two points, so x is not equal to 0
therefore
2x=y^2+y,x>0