Find scalar quantity linking two lengths The question is a GCSE maths question per the picture below. A friend asked me and I can't figure it out.. and I have a maths degree.. embarrassing! What would be a good approach to this question?
$|AP|=k\,|AB|$, $\angle NOB=\alpha$.

Let $[OBN]$ denote the area of $\triangle OBN$.

\begin{align} 
[AOM] &amp;=[AMB]= \tfrac12\,ab\sin\alpha
,\\
[OBN]&amp;=\tfrac12\cdot3a\cdot2b\,\sin\alpha=3ab\sin\alpha
,\\
[ABN]&amp;=[OBN]-2[AOM]=2\,ab\sin\alpha=4[AOM] 
,\\
[OMN]&amp;=\tfrac12[OBN]=\tfrac32 ab\sin\alpha
,\\
[AMP]&amp;=k[AMB]
,\\
[APN]&amp;=k[ABN]
,\\
[AMP]+[APN]&amp;=
[OMN]-[AOM]=ab\sin\alpha
,\\
[AMP]+[APN]&amp;=
k([AMB]+[ABN])=\tfrac52\,k\,ab\sin\alpha
,\\
k&amp;=\tfrac25
.
\end{align}
![enter image description -
By putting X just in middle, I require this area in exact two equal area.
Means where Y fall ?
Means shaded area is equal to unshaded area.

](https://i.stack.imgur.com/c6fgP.jpg)