A variable line passing through $(l, m)$ intersects the coordinate axes at the points $A$ and $B$. If the line drawn parallel to $Y$-axis through $A$ and parallel to $X$-axis through $B$ meet at $P$, then the locus of $P$ is

• Correct answer
  A

  $\frac{l}{x}+\frac{m}{y}=1$
• B

  $\frac{x}{1}+\frac{y}{m}=1$
• C

  $\frac{m}{x}+\frac{1}{y}=1$
• D

  $\frac{x}{m}+\frac{y}{1}=1$

Solution

Let $O A=a$, and $O B=b$
Then, the coordinate of a point $P$ is $(a, b)$. Equation of a variable line $\frac{x}{a}+\frac{y}{b}=1$ Also the line passes through $(l, m)$ $$ \frac{l}{a}+\frac{m}{b}=1 $$ $\therefore$ The locus of a point $P$ is $\frac{l}{x}+\frac{m}{y}=1$

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