Suppose AB is a focal chord of the parabola y^2 = 12x of length l and slope m < √3.

Question

Suppose AB is a focal chord of the parabola \(y^2 = 12x\) of length \(l\) and slope m < \(\sqrt 3\). If the distance of the chord AB from the origin is \(d\), then \(ld^2\) is equal to __________.

Answer 1

Correct answer: 108

\(2 x-\left(t-\frac{1}{t}\right) y-6=0\)

\(l=\sqrt{36\left(t+\frac{1}{t}\right)^2+9\left(t^2-\frac{1}{t^2}\right)^2}\)

\(l=3\left(t+\frac{1}{t}\right) \sqrt{4+\left(t-\frac{1}{t}\right)^2} \)

\(l=3\left(\mathrm{f}+\frac{1}{t}\right)^2 ; d=\frac{6}{\sqrt{4+\left(t-\frac{1}{t}\right)^2}} \)

\(\Rightarrow l \times d^2=3\left(t+\frac{1}{t}\right)^2 \times \frac{36}{\left(t+\frac{1}{t}\right)^2}=108\)