A variable line passing through a fixed point $P .$ The algebraic sum of the perpendiculars drawn from the poits $(2, 0), (0, 2)$ and $(1, 1)$ on the line is zero. Find the coordinates of the point $P$

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Solution

$\begin{matrix} L e t t h e v a r i a b l e l i n e b e a x + b y = 1 I t i s g i v e n t h a t t h e a e b r a i c s u m o f t h e p e r p e n d i c u l a r f r o m t h e p o i n t (2, 0), (0, 2), (1, 1) t o t h i s l i n e i s z e r o . ∴ \frac{2 a + 0 b - 1}{\sqrt{a^{2} + b^{2}}} + \frac{0 a + 2 b - 1}{\sqrt{a^{2} + b^{2}}} + \frac{a + b - 1}{\sqrt{a^{2} + b^{2}}} = 0 \Rightarrow 3 a + 3 b - 3 = 0 \Rightarrow a + b - 1 = 0 ∴ a + b = 1 s o, t h e e q u a t i o n a x + b y = 1 r e p r e s e n t s a f a m i l y o f s t r a i g h t l i n e s p a s s i n g t h r o u g h a f i x e d p o i n t . C o m p a r i n g a x + b y = 1 a n d a + b = 1 w e o b t a i n s t h a t t h e c o o r d i n a t e s o f t h e f i x e d p o i n t s a r e (1, 1) . \end{matrix}$

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