Possible length of focal chord of parabloa

Question

ain this I am not able to understand the slution Can anybody please explain me .

Answer 1

Use the property of parabola :

"The harmonic mean of segments of a focal chord is equal to semi-latus rectum"

I.e. for a parabola $y^2=4ax$,

$$\frac{1}{m}+\frac{1}{n}= \frac{2}{\underbrace{2a}_{\text{semi-latus rectum}}}$$

(In the given solution, $m$ and $n$ are length of segments of focal chord.)

Now all you need is to factorize it into the given way :

$$(m-2)(n-2)=4 \implies (m-2)=4 ~\text{and}~ (n-2)=1 ; ~~\text{or}~ (m-2)=2 ~\text{and}~ (n-2)=2$$