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$P(a, b)$ is the mid-point of a line segment between axes. Show that equation of the line is
$\frac{x}{a}+\frac{y}{b}=2$
$\frac{x}{a}+\frac{y}{b}=2$
Solution:
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Verified Answer
Let the line $A B$ makes intercepts $p, q$ on the axes.
$\therefore \quad A$ is $(p, 0)$ and $B$ is $(0, q)$
Now, $P(\mathrm{a}, \mathrm{b})$ is the mid-point of $A B$
$\begin{aligned}
&\therefore \quad \frac{\mathrm{p}+0}{2}=\mathrm{a}, \frac{0+\mathrm{q}}{2}=\mathrm{b} \\
&\therefore \quad p=2 a, q=2 b
\end{aligned}$
Intercept form of the line $A B$ is
$\frac{x}{p}+\frac{y}{q}=1 \Rightarrow \frac{x}{2 a}+\frac{y}{2 b}=1$
$\Rightarrow \frac{x}{a}+\frac{y}{b}=2$
$\therefore \quad A$ is $(p, 0)$ and $B$ is $(0, q)$
Now, $P(\mathrm{a}, \mathrm{b})$ is the mid-point of $A B$
$\begin{aligned}
&\therefore \quad \frac{\mathrm{p}+0}{2}=\mathrm{a}, \frac{0+\mathrm{q}}{2}=\mathrm{b} \\
&\therefore \quad p=2 a, q=2 b
\end{aligned}$
Intercept form of the line $A B$ is
$\frac{x}{p}+\frac{y}{q}=1 \Rightarrow \frac{x}{2 a}+\frac{y}{2 b}=1$
$\Rightarrow \frac{x}{a}+\frac{y}{b}=2$
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