Our teacher has given us a question to be solved.
What curve does the intersection points of the given lines make? A parabola, hyperbola, or none of them? (please look at the image I've posted. I am not speaking about all of the intersection points. Just the tangent curve beneath the lines)
the lines are as follows:
$y=\frac{4\sqrt{3}}{5}x+\frac{41\sqrt{3}}{10}$
$y=\frac{3\sqrt{3}}{5}x+\frac{17\sqrt{3}}{5}$
$y=\frac{2\sqrt{3}}{5}x+\frac{29\sqrt{3}}{10}$
$y=\frac{\sqrt{3}}{5}x+\frac{13\sqrt{3}}{5}$
$y=\frac{5\sqrt{3}}{2}$
$y=-\frac{\sqrt{3}}{5}x+\frac{13\sqrt{3}}{5}$
$y=-\frac{2\sqrt{3}}{5}x+\frac{29\sqrt{3}}{10}$
$y=-\frac{3\sqrt{3}}{5}x+\frac{17\sqrt{3}}{5}$
$y=-\frac{4\sqrt{3}}{5}x+\frac{41\sqrt{3}}{10}$
$y=\sqrt{3}x+5\sqrt{3}$
$y=-\sqrt{3}x+5\sqrt{3}$
my solution: I tried to find if it is a parabola first. Assuming the maximum point to be $(0,\frac{13\sqrt{3}}{5})$ (the intersection point of $4$th and $6$th lines), I found the equation to be $y=-\frac{13\sqrt{3}}{125}x^2+\frac{13\sqrt{3}}{5}$. But after plotting it in desmos, the curve didn't meet the intersection points.
Then I tried to assume $(0,\frac{5\sqrt{3}}{2})$ as the maximum; I yielded $y=-\frac{\sqrt{3}}{10}x^2+\frac{5\sqrt{3}}{2}$; but it again failed in plotting.
I then switched to hyperbola: After a long effort I found the equations to be $$\frac{(y-5\sqrt{3})^2}{(\frac{5\sqrt{3}}{2})^2}-\frac{x^2}{(\frac{5\sqrt{3}}{3})^2}=1$$ and $$\frac{(y-5\sqrt{3})^2}{(\frac{12\sqrt{3}}{5})^2}-\frac{x^2}{(\frac{5\sqrt{3}}{3})^2}=1$$ but I hink both of them are incorrect; because they are irrelevant to their asymptotes ($10$th & $11$th lines in the set of lines).
Sorry, I had to stretch the picture a little so that it better be displayed.