SATHEE: Straight Line and Pair of Straight Lines 1 Question 44

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.

$\begin{array}{cc} ∴ & \frac{2 a + 0 + c}{\sqrt{a^{2} + b^{2}}} + \frac{0 + 2 b + c}{\sqrt{a^{2} + b^{2}}} + \frac{a + b + c}{\sqrt{a^{2} + b^{2}}} = 0 \\ \Rightarrow & 3 a + 3 b + 3 c = 0 \\ \Rightarrow & a + b + c = 0 \end{array}$

From Eqs. (i) and (ii) $a x + b y + c = 0$ always passes through a fixed point $(1, 1)$ .

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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… .

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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… .