A variable line $\frac{x}{a}+\frac{y}{b}=1$ is such that $a+b=4$. The locus of the mid point of the portion of the line intercepted between the axes is

• A

  $x+y=4$
• B

  $x+y=8$
• C

  $x+y=1$
• Correct answer
  D

  $x+y=2$

Solution

Let the coordinate of mid point of $A B$ is $\left(x_{1}, y_{1}\right)$.

$$ \begin{aligned} &\Rightarrow \quad \mathrm{x}_{1}=\frac{\mathrm{a}+0}{2}, \mathrm{y}_{1}=\frac{0+\mathrm{b}}{2} \\ &\therefore \quad \mathrm{a}=2 \mathrm{x}_{1}, \mathrm{~b}=2 \mathrm{y}_{1} \\ &\text { Given, } \quad \mathrm{a}_{1}+2 \mathrm{~b}_{1}=4 \\ &\therefore \quad \mathrm{x}_{1}+\mathrm{y}_{1}=2 \end{aligned} $$ Hence, the locus of the mid point is $$ x+y=2 $$

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