The chord joining the points P(ap2,2ap) and Q(aq2,2aq)
y=mx+b
slope=m=aq2−ap22aq−2ap=q+p2,a=0,q=±p,
y=q+p2x+b Point P(ap2,2ap)
2ap=q+p2(ap2)+b
b=q+p2(apq+ap2−ap2)
b=q+p2apq
y=q+p2x+q+p2apq Point (a,0)
0=q+p2a+q+p2apq
pq=−1Therefore if the chord joining the points P(ap2 , 2ap), Q(aq2 , 2aq) on the parabola y2= 4ax passes through (a, 0), then pq = −1.
The tangent at P(ap2,2ap)
y2=4ax
2yy′=4a
y′=y2a
slope=2ap2a=p1
y−2ap=p1(x−ap2)
y=p1x+apThe tangent at P meets the line through Q parallel to the axis of the parabola at R
y=2aq
p1x+ap=2aq
x=2apq−ap2
R(2apq−ap2,2aq)
PR:2x1+x2=2ap2+2apq−ap2=apq
2y1+y2=22ap+2aq=ap+aq The middle of PR is the point (apq,ap+aq).
If pq=−1, then
2x1+x2=apq=−aThe middle of PR is the point (−a,ap+aq).
Then the line x+a=0 bisects PR.
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