Given that the chord ab of parabola y ^ 2 = 2x passes through the fixed point (- 2,0), the trajectory equation of the midpoint of the chord AB is obtained

Given that the chord ab of parabola y ^ 2 = 2x passes through the fixed point (- 2,0), the trajectory equation of the midpoint of the chord AB is obtained

If the slope of AB exists
Let the slope of AB be K
y=k(x+2)=kx+2k
So (KX + 2K) & sup2; = 2x
k²x²+(4k²-2)x+4k²=0
x1+x2=-(4k²-2)/k²
y=kx+2k
So Y1 + y2 = K (x1 + x2) + 4K = 2 / K
For the middle point, x = (x1 + x2) / 2, y = (Y1 + Y2) / 2
So y / x = (2 / k) / [- (4K & sup2; - 2) / K & sup2;] = K / (1-2k & sup2;)
y=k(x+2)
So k = Y / (x + 2)
Substituting
y(1-2k²)=kx
That is y (X & sup2; + 4x + 4-2y & sup2;) = XY (x + 2)
x²+4x+4-2y²=x²+2x
y²=x+2
If there is an intersection, then K & sup2; X & sup2; + (4K & sup2; - 2) x + 4K & sup2; = 0 has a solution
Discriminant = 16K ^ 4-16k & sup2; + 4-16k & sup2; > = 0
k²

Given that the parabola y ^ 2 = 2x, make a straight line intersecting the parabola at two points a and B through the point Q (2,1), and try to find the trajectory equation of the midpoint of the chord ab
I can take y = K (X-2) + 1 into y ^ 2 = 2x and get k ^ 2 (x-1) ^ 2 + 4K (x-1) + (4-2x) = 0. I don't know what to do after using Veda's theorem. I'm not good at conic curve. I hope someone can help me,
But it's better to have some calculation,

Tip: the point difference method is the simplest
Let a (x1, Y1), B (X2, Y2), midpoint P (x0, Y0)
So Y1 ^ 2 = 2x1
y2^2=2x2
Subtraction results in:
(y1-y2)(y1+y2)=2(x1-x2)
The slope of line AB is k = (y1-y2) / (x1-x2) = 2 / (Y1 + Y2)
Easy to know: Y1 + y2 = 2y0
That is, k = 1 / Y0
And the slope k = (y0-1) / (x0-2)
So (y0-1) / (x0-2) = 1 / Y0
So x0 = Y0 ^ 2-y0 + 2
So the trajectory equation: x = y ^ 2-y + 2

It is known that parabola X & # 178; = 4Y, circle X & # 178; + Y & # 178; = 1
Let P (a, b) be a point (a > 2) on the parabola, and make two tangent lines of the circle through P, which respectively intersect with the X axis at two points a and B. if the midpoint of the line AB is (- 4 / 15,0), find the value of the real number a

Obviously, let a (m, 0), PA's equation: (y-0) / (x-m) = (a-178) / 4-0) / (A-M) a-178; x-4 (A-M) y-a-178; m = 0 circle x-178; + y-178; = 1 circle center as origin o, radius r = 1AP tangent to circle, then O and ap