question_answer

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)  \[x+y=1\]

D)  \[x+y=2\]

Correct Answer: D

Solution :

Let the coordinate of mid point of AB is \[({{x}_{1}},{{y}_{1}})\].                 \[\therefore \]  \[{{x}_{1}}=\frac{a+0}{2},{{y}_{1}}=\frac{0+b}{2}\]                 \[\Rightarrow \]               \[a=2{{x}_{1}},b=2{{y}_{1}}\] Given,   \[a+b=4\] \[\therefore \]  \[2{{x}_{1}}+2{{y}_{1}}=4\] \[\Rightarrow \]               \[{{x}_{1}}+{{y}_{1}}=2\] Hence, the locus of the mid point is                 \[x+y=2\]