Straight Line and Pair of Straight Lines 1 Question 44
44. Let the algebraic sum of the perpendicular distance from the points $(2,0),(0,2)$ and $(1,1)$ to a variable straight line be zero, then the line passes through a fixed point whose coordinates are… .
(1991, 2M)
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Solution:
- Let the variable straight line be $a x+b y+c=0$
where, algebraic sum of perpendiculars from $(2,0),(0,2)$ and $(1,1)$ is zero.
$$ \begin{array}{cc} \therefore & \frac{2 a+0+c}{\sqrt{a^{2}+b^{2}}}+\frac{0+2 b+c}{\sqrt{a^{2}+b^{2}}}+\frac{a+b+c}{\sqrt{a^{2}+b^{2}}}=0 \\ \Rightarrow & 3 a+3 b+3 c=0 \\ \Rightarrow & a+b+c=0 \end{array} $$
From Eqs. (i) and (ii) $a x+b y+c=0$ always passes through a fixed point $(1,1)$.
