Let |KF|=p, then the equation of this parabola can be obtained as? y2 = 2px(p > 0)；
(ii) This proposition is true, which is proved as follows:
As shown in Figure 2, let the midpoint of PQ be M, and the projections of M on the parabolic directrix L are A, B and D respectively.
Pq is the chord of a parabola passing through the focal point f,
∴|PF|=|PA|, |QF|=|QB|, and |MD| is the center line of trapezoidal APQB.
∴|md= 12(|pa|+|qb|)= 12(|pf|+|qf|)=|pq|2.
∵M is the center of the circle with PQ as the diameter,
The circle m is tangent to l.
(3) Choose Elliptic Analogy (2) Write the following proposition:
The straight line passing through the focus f of the ellipse intersects the ellipse at two points P and Q, and then the circle with diameter PQ is separated from the corresponding directrix L of the ellipse.
This proposition is true, which is proved as follows:
It is proved that if the midpoint of PQ is m and the eccentricity of ellipse is e,
Then the projections of 0 < e < 1, p, q and m on the corresponding directrix l are a, b, d respectively,
∵| pf | pa = e, ∴| pa | = | pf | e, in the same way | QB | = | qf | e.
∫MD is the center line of trapezoidal APQB,
∴|md|=|pa|+|qb|2= 12(|pf|e+|qf|e)=|pq|2e>|pq|2，
∴: the circle m is separated from the directrix L.
Choose hyperbolic analogy (Ⅱ) to write the proposition as follows:
The straight line passing through the focal point f of hyperbola intersects hyperbola at p and q, and then the circle with diameter PQ intersects the corresponding directrix L of hyperbola.
This proposition is true, which is proved as follows:
It is proved that if the midpoint of PQ is m and the eccentricity of ellipse is e,
Then the projections of e > 1, p, q and m on the corresponding directrix l are a, b and d, respectively.
∵| pf | pa = e, ∴| pa | = | pf | e, in the same way | QB | = | qf | e.
∫MD is the center line of trapezoidal APQB,
∴|md|=|pa|+|qb|2= 12(|pf|e+|qf|e)=|pq|2e