If the midpoint of the chord PQ of the parabola y & # 178; = 2px (P > 0) is m (x0, Y0) (Y0 ≠ 0), then the slope of the straight line PQ is

If the midpoint of the chord PQ of the parabola y & # 178; = 2px (P > 0) is m (x0, Y0) (Y0 ≠ 0), then the slope of the straight line PQ is

(yP)^2-(yQ)^2=2pxP-2pxQ
(yP+yQ)*(yP-yQ)=2p(xP-xQ)
2y0*(yP-yQ)/(xP-xQ)=2p
k(PQ)=(yP-yQ)/(xP-xQ)=p/y0

On the parabola y = x ^ 2, there is a moving chord ab | ab | = 2. The trajectory equation of the midpoint m of the moving chord is obtained
As above

Let m (x, y), a (x1, Y1), B (X2, Y2), the equation of line AB be y = KX + B. from y = KX + B, y = x ∧ 2, we get x ∧ 2-kx-b = 0. From the violation theorem, we get x 1 + X2 = k, X 1x2 = - B, Y 1 + y 2 = kx1 + B + kx2 + B = K (x1 + x2) + 2B = k ∧ 2 + 2B. From the chord length formula ∧ ab ∧ = √ (1 + K ∧ 2) ∧ x1-x2