If the centroid of the triangle $A B C$ is at origin and algebraic sum of the lengths of perpendiculars from the vertices of the triangle on the line $L$ , is equal to $1$ , then prove that sum of the co-ordinate axes is $9$ .

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Solution

Given $\frac{Σ x_{i}}{3} = 0, \frac{Σ y_{i}}{3} = 0, i = 1, 2, 3$ (1)
and $p_{1} + p_{2} + p_{3} = 0 ⟹ \frac{a Σ x_{i} + b Σ y_{i} + 3 c}{\sqrt{a^{2} + b^{2}}} = 1$
$∴ 9 c^{2} = a^{2} + b^{2}$ by (1) (2)
Now sum of the square of the intercepts made by the line on co-ordinate axes is
${(- \frac{a}{c})}^{2} + {(\frac{- b}{c})}^{2} = \frac{a^{2} + b^{2}}{c^{2}} = 9$ by (2)

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If the centroid of the triangle ABC is at origin and algebraic sum of the lengths of perpendiculars from the vertices of the triangle on the line L, is equal to 1, then prove that sum of the co-ordinate axes is 9.

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