Question

If $l$ and $l^{\prime }$ are the lengths of segment of focal chord of a parabola $y^2=4ax$, then prove that $\frac{1}{l}+\frac{1}{l^{\prime }}=\frac{1}{a}$

Answer 1

We use the parametric form of the parabola $y^2=4ax$ which is $(a{t}^2,2at)$.

By the property of focal chords, if one end of a focal chord is $A=(a{t}_1^2,2a{t}_1)$, then the other end is

$B=(\frac{a}{{t_1}^2},-\frac{2a}{t_1})$.

The focus is $S=(a,0)$.

Then, the lengths of the segments of the focal chords are found by:

$l=AS=\sqrt{a^2(t_1^2-1{)}^2+4a^2{t}_1^2}=a(t_1^2+1)$

$l^{\prime }=BS=\sqrt{a^2{(\frac{1}{{t_1}^2}-1)}^2+4\frac{a^2}{t_1^2}}=a(\frac{1}{{t_1}^2}+1)$

Now,

$\frac{1}{l}+\frac{1}{l^{\prime }}=\frac{1}{a(t_1^2+1)}+\frac{t_1^2}{a(t_1^2+1)}=\frac{1}{a}$

Hence Proved.