An equation of a line whose segment between the coordinate axes is divided by the point $\left(\frac{1}{2}, \frac{1}{3}\right)$ in the ratio $2: 3$, is

• A

  $6 x+9 y=5$
• B

  $9 x+6 y=5$
• Correct answer
  C

  $4 x+9 y=5$
• D

  $9 x+4 y=5$

Solution

Let $A$ and $B$ be the points on the coordinate axes, where the line intersect.
Since, $P$ divides the line segment $B A$ in the ratio $2: 3$. Then, the coordinates of point $P$ will be $$ \left(\frac{2 a+0}{2+3}, \frac{0+3 b}{2+3}\right) \text { i.e. }\left(\frac{2 a}{5}, \frac{3 b}{5}\right) . $$ But coordinates of $P$ are $\left(\frac{1}{2}, \frac{1}{3}\right)$. $$ \begin{aligned} & \therefore \frac{2 a}{5}=\frac{1}{2} \text { and } \frac{3 b}{5}=\frac{1}{3} \Rightarrow a=\frac{5}{4} \\ & \text { and } \\ & b=\frac{5}{9} \\ & \end{aligned} $$ So, equation of line will be $$ \begin{gathered} \frac{x}{\frac{5}{4}}+\frac{y}{\frac{5}{9}}=1 \\ 4 x+9 y=5 \end{gathered} $$

Hence, the correct answer is option: 'C'.