If the line x = a intersects the
parabolas y = x2, y = x2 + 1, y = x2 + 2, y = x2 – 1, y = x2 – 2 at P0, P1, P2, P–1, P–2 etc ,
respectively, then dy
dx  at these points equals 2a This indicates that the tangents to the
curves at these points are parallel Thus, 
2
C
2
C
F ( )
x dx
x
x
=
+
=
∫
(say), implies that
292
MATHEMATICS
the tangents to all the curves y = F C (x), C ∈ R, at the points of intersection of the
curves by the line x = a, (a ∈ R), are parallel