Question

If the algebraic sum of the perpendiculars drawn from the points $(2,0),(0,2),(1,1)$ to a variable line is zero, then the line will always pass through a fixed point whose co-ordinates are

• A. $(1,1)$
• B. $(2,2)$
• C. $(3,3)$
• D. $(0,0)$

Answer 1

The correct option is A $(1,1)$

Let the variable line be
$ax+by=1...\left(1\right)$
If it is given that the algebraic sum of perpendiculars drawn from the points $(2,0),(0,2),(1,1)$ is zero.
$\therefore \frac{2a-1}{\sqrt{a^2+b^2}}+\frac{2b-1}{\sqrt{a^2+b^2}}+\frac{a+b-1}{\sqrt{a^2+b^2}}=0$
$\Rightarrow 3a+3b-3=0$
$\Rightarrow a+b=1...\left(2\right)$
Comparing $\left(1\right)$ and $\left(2\right),$ we get
$(x,y)=(1,1)$
So, variable line always passes through fixed point $(1,1)$