Decide if a point is inside the shape made by a fixed-area isosceles triangle as its vertex slides down the y-axis

Question

I have an isosceles triangle with a vertex $(0,h)$ that lies on the y-axis and the base defined by two points $(-a,0)$ and $(a,0)$.

The vertex slides down the $y$-axis (the value of $h$ decreases) while keeping the area constant like so: triangle with sliding vertex

Given a point $P$, how do I decide if $P$ is ever inside the triangle?

Answer 1

Here's an easy way to think about it. You want the the lines passing through $(0,hk)$ and $(a/k,0)$ as $k$ varies between $0$ and $1$. You can check that this line has equation $$y=hk(1 - \frac ka x).$$ So, given a point $(x_0,y_0)$, you want to know if $y_0<hk(1-\frac kax_0)$ for some value of $k$ between $0$ and $1$. So, fixing $x_0$ and varying $k$ between $0$ and $1$, what's the greatest the quantity $hk(1-\frac kax_0)=h(k-\frac{x_0}ak^2)$ can be?