A rod slides with its ends on two coordinate axes. A point $P$ divides the rod in the ratio $1:2$. Find the locus of $P$.
Let coordinate of $P$ is $(h, k)$. Let coordinate of $A$ is $(0, a)$ and coordinate of $B$ is $(b, 0)$. Now, $\mathrm{AP}:\mathrm{PB} = 1:2$ $$h = b/3, \quad k = 2a/3$$ Now, length of the rod remains constant: $$\sqrt{a^2 + b^2} = c \Rightarrow (3k/2)^2 + (3h)^2 = c^2 \Rightarrow 4h^2 + k^2 = c_1^2$$ The locus of $P$ is $4x^2 + y^2 = c_1^2$. (B)
A rod slides with its ends on two coordinate axes. A point $P$ divides the rod in the ratio $1:2$. Find the locus of $P$.