The line $6x+8y=48$ intersects the $x$-axis in point $A$ and the $y$-axis in point $B$. A line $L$ bisects the area and the perimeter of the triangle $OAB$,where $O$ is origin.
Find possible equation(s) of $L$.
My Attempt
I feel there can be three lines but how to proceed.
On
Let the line $L$ be $y= k x +b$ and it intersects with $6x+8y=48$ at $(p,q)$, where
$$p= \frac{24-4b}{3+4k },\>\>\>\>\> q= \frac{24k+3b}{3+4k } $$
Given the equal area and perimeter, establish the equations below
$$(6-b)p=bp+8q$$ $$(6-b)+\sqrt{p^2+(6-q)^2} = b+8+\sqrt{(8-p)^2+q^2}$$
Solve the system of equations above to obtain $b=\sqrt6$ and $k=1-\sqrt{\frac32}$. Thus, the bisecting line $L$ is
$$y= \left(1-\sqrt{\frac32}\right)x+\sqrt6$$
(Note that above equations corresponds to $L$ intersecting the vertical leg and the hypotenuse; other configurations do not yield valid solutions.)
There are three conditions possible :
1. $L$ intersects $OA$ and $AB$.
Area of $\displaystyle\triangle ACD=\frac{1}{2}(pq)\sin37^\circ=12\implies pq=40$. Also, $p+q=12$ for perimeter bisection.
Solve for $p,q$.
Do the same for other two cases
2. $L$ intersects $OB$ and $AB$.
3. $L$ intersects $OA$ and $OB$.
As corrected by @Moko19, also check the other three cases, too which I didn't mentioned in the diagram.