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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
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The correct answer is:
$x+y=2$
We have, equation of line $\frac{x}{a}+\frac{y}{b}=1$.
Let $P(h, k)$ be the required point.
Now, coordinates $A$ and $B$ are $(a, 0)$ and $(0, b)$ respectively. Since, $P$ is mid-point of $A B$.
Therefore,
$(h, k)=\left(\frac{a+0}{2}, \frac{0+b}{2}\right)$
$\Rightarrow \quad h=\frac{a}{2}$ and $k=\frac{b}{2}$
$\Rightarrow \quad a=2 h$ and $b=2 k$
Now, it is given that
$$
\begin{aligned}
& a+b=4 \\
\Rightarrow & 2 h+2 k=4 \\
\Rightarrow & h+k=2
\end{aligned}
$$
So, locus of $P$ is $x+y=2$
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