Given the ellipse E $4x²+9y²=36$ and a point $P=(4,7)$. Let $Q=(x,y)$ point of the ellipse and $R$ a simetric point of Q respect $P$. Find the geometrical place of $R$.
Ok, i think that R belong to a parabola, but i don't know how to find that equation. Any suggestion?
To each point $Q\in E$ you assign a symmetric point $R$ - so you create a symmetric image $F$ of the whole ellipse $E$. The geometrical place of $Q$ is $E$ (the original ellipse), the geometrical place of $R$ is $F$, which is also an ellipse (not a parabola).
Now if $Q=(x,y)$, let's denote $R=(a,b)$ and then $$(a,b)=R=Q+2\cdot (P-Q)=\left(x+2\cdot(4-x), y+2\cdot(7-y)\right)=(8-x, 14-y)$$ $$a=8-x,\ b=14-y$$ $$x=8-a,\ y=14-b$$ We can substitute this to the equation for ellipse $E$, and we get an equation for the geometrical place of points $R=(a,b)$, given in terms of $a,b$: $$4\cdot{(8-a)}^2+9\cdot{(14-b)}^2=36,$$ which is an equation for ellipse, earlier denoted as $F$.