Question

If b and c are lengths of the segments of any focal chord of the parabola y² = 4ax, then write the length of its latus-rectum.

Answer 1

Let S (a, 0) be the focus of the given parabola.

Let the end points of the focal chord be $P\left(a{t}^2,2at\right)\mathrm{and}Q\left(\frac{a}{t^2},\frac{-2a}{t}\right)$.

SP and SQ are segments of the focal chord with lengths b and c, respectively.

∴ SP = b, SQ = c

$$\begin{gathered} \mathrm{Also},SP=\sqrt{\left(a-a{t}^2\right)+4a^2{t}^2}=a\left(1+t^2\right) \\ \mathrm{And},SQ=\sqrt{\left(a-\frac{a}{t^2}\right)+\frac{4a^2}{t^2}}=a\left(1+\frac{1}{t^2}\right) \end{gathered}$$

Now, we have:
$\frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a\left(1+t^2\right)}+\frac{t^2}{a\left(1+t^2\right)}=\frac{1}{a}$

$$\begin{gathered} \Rightarrow \frac{1}{b}+\frac{1}{c}=\frac{1}{a} \\ \Rightarrow \frac{b+c}{bc}=\frac{1}{a} \\ \Rightarrow a=\frac{bc}{b+c} \end{gathered}$$

∴ Length of the latus rectum = 4a = $\frac{4bc}{b+c}$