The ends of a rod  of length $l$ move on the coordinate axes . The locus of the point on the rod which divides it in the ratio $1 : 2$is 

Let the rod meets the axes in $A\left(a ,0\right) and B\left(0 ,b\right)$

$$\begin{gathered} given AB=l \\ \Rightarrow \sqrt{a^2+b^2}=l\Rightarrow a^2+b^2=l^2------\left(1\right) \end{gathered}$$

Let $P\left(x , y\right)$ be the point on locus 

$$\begin{gathered} \Rightarrow P divides AB in the ratio 1:2 \\ \Rightarrow \left(x ,y\right)=\left(\frac{1\left(0\right)+2\left(a\right)}{1+2} , \frac{1\left(b\right)+2\left(0\right)}{1+2}\right)\Rightarrow \left(\frac{2a}{3} ,\frac{b}{3}\right)=\left(x ,y\right) \\ \Rightarrow a=\frac{3x}{2} and b=3y \\ \therefore from \left(1\right) \\ {\left(\frac{3x}{2}\right)}^2+{\left(3y\right)}^2=l^2 \\ \Rightarrow 9x^2+36y^2=4l^2 \\ \Rightarrow \end{gathered}$$