Question

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

• A. $9x^2+36y^2=4l^2$
• B. $9x^2+36y^2=16l^2$
• C. $144x^2+16y^2=9l^2$
• D. $4x^2+36y^2=9l^2$

Answer 1

Answer: (C)

$144x^2+16y^2=9l^2$

Length of a rod is $l$ with ends $(a,0)$ and $(0,b)$

Now rod and axes form a right angle triangle

$⟹a^2+b^2=l^2$ ............ $\left(i\right)$

Let $(h,k)$ be the point which divides a rod in the ratio $1:3$

$\therefore h=\frac{(a\times 1)+(0\times 3)}{4}$ and $k=\frac{(3\times b)+(1\times 0)}{4}$

$⟹4h=a\text{and}\frac{4k}{3}=b$

$⟹16h^2+\frac{16k^2}{9}=a^2+b^2$ ............ Squaring and adding above eq${}^{ns}$
$⟹16h^2+\frac{16k^2}{9}=l^2$ ............. From $\left(i\right)$
$⟹144h^2+16k^2=9l^2$ ............. Multiplying by $9$
$\therefore 144x^2+16y^2=9l^2$ is the required locus of $(h,k)$

Hence, option C is correct.