First slide

Equation of line in Straight lines

Question

Let the algebraic sum of the perpendicular distances from the points A(2, 0), B(0, 2), C(1, 1) to a variable line be zero. Then, all such lines

Moderate

A

are concurrent
B

pass through the fixed point (1, 1)
C

touch some fixed circle
D

pass through the centroid of ΔABC

Solution

Let the variable line be ax + by + c = 0 (1)
Given, $\frac{[(2 a + c) + (2 b + c) + (a + b + c)]}{\sqrt{a^{2} + b^{2} + c^{2}}} = 0$
⇒ 3a + 3b + 3c = 0 or a + b + c = 0.
So, the equation of the line becomes
ax + by – a – b = 0
or, a (x – 1) + b (y – 1) = 0.
⇒ the line passes through the point of intersection of lines
x – 1 = 0 and y – 1 = 0, i.e., the fixed point (1, 1).
So, all such lines are concurrent. Also, (1, 1) is the centroid
of the ΔABC.

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