$P(a, b)$ is the mid-point of a line segment between the axes. Show that the equation of the line is $\frac{x}{a} + \frac{y}{b} = 2$.
(N/A) Let $AB$ be the line segment between the axes and let $P(a, b)$ be its mid-point.
Let the coordinates of $A$ and $B$ be $(0, y_0)$ and $(x_0, 0)$ respectively.
Since $P(a, b)$ is the mid-point of $AB$,we have:
$\left(\frac{0 + x_0}{2}, \frac{y_0 + 0}{2}\right) = (a, b)$
$\Rightarrow \left(\frac{x_0}{2}, \frac{y_0}{2}\right) = (a, b)$
$\Rightarrow \frac{x_0}{2} = a$ and $\frac{y_0}{2} = b$
$\therefore x_0 = 2a$ and $y_0 = 2b$
Thus,the coordinates of $A$ and $B$ are $(0, 2b)$ and $(2a, 0)$ respectively.
The equation of the line passing through points $(0, 2b)$ and $(2a, 0)$ using the intercept form $\frac{x}{X} + \frac{y}{Y} = 1$ where $X=2a$ and $Y=2b$ is:
$\frac{x}{2a} + \frac{y}{2b} = 1$
Multiplying both sides by $2$,we get:
$\frac{x}{a} + \frac{y}{b} = 2$
Let the coordinates of $A$ and $B$ be $(0, y_0)$ and $(x_0, 0)$ respectively.
Since $P(a, b)$ is the mid-point of $AB$,we have:
$\left(\frac{0 + x_0}{2}, \frac{y_0 + 0}{2}\right) = (a, b)$
$\Rightarrow \left(\frac{x_0}{2}, \frac{y_0}{2}\right) = (a, b)$
$\Rightarrow \frac{x_0}{2} = a$ and $\frac{y_0}{2} = b$
$\therefore x_0 = 2a$ and $y_0 = 2b$
Thus,the coordinates of $A$ and $B$ are $(0, 2b)$ and $(2a, 0)$ respectively.
The equation of the line passing through points $(0, 2b)$ and $(2a, 0)$ using the intercept form $\frac{x}{X} + \frac{y}{Y} = 1$ where $X=2a$ and $Y=2b$ is:
$\frac{x}{2a} + \frac{y}{2b} = 1$
Multiplying both sides by $2$,we get:
$\frac{x}{a} + \frac{y}{b} = 2$
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