Say point A (x1, y1) = (ap12,2ap1) , and
point B (x2, y3) = (ap22,2ap2) coordinates of the ends of a focal chord of the
parabola y2=4ax
Then O, A and B are collinear, where O is the vertex.
Therefore, Slope of OA = Slope of OB
Or, (ap12−a)2ap1=(ap22−a)2ap2
On solving,, we get
p1p22−p1=p12p2−p2
p1p2(p2−p1)=−(p2−p1)
Or
p1p2=−1
Now
x1x2=ap12.ap22
Or
x1x2=a2(1)=a2
Therefore,
x1x2=a2(Proved)
Now
y1y2=2ap12ap2=4a2p1p2=−4a2
Therefore,
y1y2=−4a2(Proved)
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