A variable line passes through a fixed point P. The algebraic sum… - Vidyadip

Question

A variable line passes through a fixed point $P.$ The algebraic sum of the perpendiculars drawn from the points $(2, 0), (0, 2)$ and $(1, 1)$ on the line is zero. Find the coordinates of the point $P.$
$[$Hint: Let the slope of the line be m. Then the equation of the line passing through the fixed point $P (x_1, y_1)$ is $y - y_1 = m (x - x_1)$. Taking the algebraic sum of perpendicular distances equal to zero$,$ we get $y - 1 = m (x - 1).$ Thus $(x_1, y_1)\ is\ (1, 1).]$

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Answer

Let the variable line throught the fixed point $P$ is $ax + by + c = 0 .....(i)$
Perpendicular distance from $\text{A}(2,0)=\frac{2\text{a}+0+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}$
Perpendicular distance from $\text{A}(0,2)=\frac{0+2\text{b}+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}$
Perpendicular distance from $\text{A}(1, 1)=\frac{\text{a}+\text{b}+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}$
According to the question$,$ we have
$\Rightarrow \frac{2\text{a}+0+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}+\frac{0+2\text{b}+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}+\frac{\text{a}+\text{b}+\text{c}}{\sqrt{\text{a}^2+\text{b}^2}}=0$
$\Rightarrow 3a + 3b + 3c = 0 or a + b + c = 0 .....(ii)$
From $(i)$ and $(ii),$ variable line passes through the fixed point $(1, 1).$

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