Pure Geometry

(c) "hence" part :

From (b),
x coordinate of R = a[ (p+q)^2 +2 – pq ] = a(p+q)^2
y coordinate of R = apq(p+q) = 2a(p+q)
Consider the equation y^2 = 4ax,
L.S. = [ 2a(p+q) ]^2 = 4a^2(p+q)^2
R.S. = 4a[a(p+q)^2] = 4a^2(p+q)^2
L.S. = R.S.
R lies on the parabola.


(d) :
Consider
p + q = s
pq = 2
p, q are roots of x^2 + sx + 2 =0
The discriminant = s^2 - 4(1)(2) = s^2 - 8 > 0 (given)

We can find distinct p, q such that p + q = s and pq = 2. By (b) and (c), the distinct normals to the parabola at (ap^2, 2ap), (aq^2, 2aq) and S pass through S.

2009-03-29 01:25:28 補充：
In(d), x^2 + sx + 2 = 0 should be x^2 - sx + 2 = 0