Let M(t, -p) be any point on a quasi-straight line, the two tangents MA and MB passing through m are parabolas, and A and B are tangent points.
Since a and b are on parabola, let A (2pm, pm 2) and B (2pn, pn 2) (m ≠ n).
Y = x 2/(4p) and y' = x/(2p) from x 2 = 4py.
At a, the tangent slope k=m and the tangent equation is MX-y-PM 2 = 0.
It passes M(t, -p), mt+p-pm 2 = 0.
That is, PM 2-TM-P = 0 (1).
At point b, the tangent slope k=n and the tangent equation is NX-y-pn 2 = 0.
If it passes through M(t, -p), nt+p-pn 2 = 0.
That is, pn 2-TN-p = 0 (2)
M is obtained from (1)(2), and n is the two roots of the equation z 2-tz-p = 0.
M+n=t/p，mn=- 1 (3)。
From a (2pm, pm 2), b (2pn, pn 2) (m ≠ n), the equation of straight line AB can be obtained as follows.
(m+n)x-2y-2pmn=0
Substitute (3) to get (t/p)x-2y+2p=0.
That is, tx-2p(y-p)=0.
A straight line always passes through F(0, p)?
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Extended data:
The description of parabola includes a point (focus) and a line (directrix). The focus is not on alignment. A parabola is the locus of a point on a plane equidistant from the directrix and the focus. Another description of parabola is that as a conical section, it is formed by the intersection of a conical surface and a plane parallel to the generatrix of the cone. The third description is algebra.
Parabolas have the property that if they are made of materials that reflect light, the light that propagates parallel to the symmetry axis of the parabola and hits its concave surface is reflected to its focus, regardless of where the parabola is reflected. On the contrary, the light generated from the point light source at the focus is reflected as a parallel ("collimated") beam, so that the parabola is parallel to the axis of symmetry. Sound and other forms of energy will have the same effect. This reflection property is the basis of many practical applications of parabola.
Baidu encyclopedia-parabola