Q.A rod of length $12$ cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point $P$ on the rod, which is $3$ cm from the end in contact with the $x$-axis.

Concept: Locus of a point on a moving rod (intercept form of a line).

Let the rod $AB$ have $A$ on the $x$-axis and $B$ on the $y$-axis.

Let $A = (a, 0)$ and $B = (0, b)$. The length $AB = 12$, so

Point $P$ is $3$ cm from $A$ (the end on the $x$-axis), so $AP = 3$ and $PB = 9$.

Using the section formula (internal division in ratio $AP : PB = 3 : 9 = 1 : 3$), the coordinates of $P$ are

Let $P = (x, y)$. Then $a = \frac{4x}{3}$ and $b = 4y$.

Substitute into $a^2 + b^2 = 144$: …

Solution

Let the rod be $AB$ with end $A=(a,0)$ on the $x$-axis and end $B=(0,b)$ on the $y$-axis. Since the rod has length $12$:

Point $P$ is $3$ cm from $A$ (the end on the $x$-axis), so $AP = 3$ and $PB = 9$, i.e. $P$ divides $AB$ from $A$ in the ratio $3:9 = 1:3$. By the section formula,

So if $P=(x,y)$, then $x = \dfrac{3a}{4} \Rightarrow a = \dfrac{4x}{3}$ and $y = \dfrac{b}{4} \Rightarrow b = 4y$.

Substituting into $(1)$: …