Question

L is a variable line such that the algebraic sum of the distances of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero. The line L will always pass through

पर्याय

• (1, 1)

• (2, 1)

• (1, 2)

• none of these

Answer 1

(1,1)
Let ax + by + c = 0 be the variable line. It is given that the algebraic sum of the distances
of the points (1, 1), (2, 0) and (0, 2) from the line is equal to zero.

$$\therefore \frac{a+b+c}{\sqrt{a^2+b^2}}+\frac{2a+0+c}{\sqrt{a^2+b^2}}+\frac{0+2b+c}{\sqrt{a^2+b^2}}=0$$

$$\Rightarrow 3a+3b+3c=0$$

$$\Rightarrow a+b+c=0$$

Substituting c = $$-$$a $$-$$b in ax + by + c = 0, we get:

$$ax+by-a-b=0$$

$$\Rightarrow a(x-1)+b(y-1)=0$$

$$\Rightarrow (x-1)+\frac{b}{a}(y-1)=0$$

This line is of the form

$$L_1+\lambda L_2=0$$,  which passes through the intersection of  $$L_1=0\text{ and }{L}_2=0,$$  i.e.

x $$-$$ 1 = 0 and y $$-$$ 1 = 0.

$$\Rightarrow$$ x = 1, y = 1