Show that the semi latus rectum of the parabola y^{2} = 4ax is a harmonic mean between the segment of any focal chord.

Equation of the given parabola is *y*^{2} = 4*ax. *

Coordinates of focus = S(*a*, 0)

Let and be the end point of the focal chord of the given parabola.

∴* t*_{1} *t*_{2} = – 1 ...(1)

Length of the semi latus rectum of the given parabola = 2*a*

Let SP and SQ be the segment of the focal chord.

Similarly,

⇒ SP, 2*a* and SQ are in H.P.

Thus, the semi latus rectum of the given parabola is the harmonic mean between the segment of the local chord.

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