If a straight line be such that algebraic sum of the perpendiculars drawn from the points $A_{1}, A_{2}$ _____, $A_{n}$ is zero, then prove that the straight line passes through a fixed point whose co-ordinate is the centroid $G$ of the given points.

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Solution

Let the straight line be $a x + b y + c = 0$ such that $p_{1} + p_{2} + p_{3} +$ ____ $+ p_{n} = 0$
$∴ P_{1} = \frac{a x_{1} + b y_{1} + c}{\sqrt{(a^{2} + b^{2})}}, p_{2} \frac{a x_{2} + b y + c}{\sqrt{(a^{2} + b^{2})}}$
$∴ p_{1} + p_{2} + p_{3} +$ ____ $p_{n} = 0$
$⟹ \frac{1}{\sqrt{a^{2} + b^{2}}} [a Σ x_{i} + b Σ y_{i} + c c] = 0$
or $a \frac{Σ x_{i}}{n} + \frac{Σ y_{i}}{n} + c = 0$
Above shows that given line passes through the point $(\frac{Σ x_{i}}{n}, \frac{Σ y_{i}}{n})$ , i.e. centroid of the given points.

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(I.I.T 1991)

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