$P (a, b)$ is the mid-point of a line segment between axes. Show that equation of the line is $\frac{x}{a} + \frac{y}{b} = 2$

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Solution

Let a line intersect the $x - y$ plane at $A (p, 0)$ and $B (0, q)$
We know that the mid point of line
$A B ⟶ p (a, b) = (\frac{p + 0}{2}, \frac{0 + q}{2}) (\frac{p}{2}, \frac{q}{2}) a = \frac{p}{2} b = \frac{q}{2} 2 a = p 2 b = q$
$∴$ Now, $A (2 a, 0) & B (0, 2 b)$
$∴$ Equation of line passing through
$(2 a, 0) (y - 0) = \frac{2 b - 0}{0 - 2 a} (x - 2 a) y = - \frac{b}{a} (x - 2 a) a y = - b x + 2 a b a y + b x = 2 a b$
Dividing by at
$\frac{x}{a} + \frac{y}{b} = \frac{2 a b}{a b} \frac{x}{a} + \frac{y}{b} = 2$

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