Question

A variable line passes through a fixed point $P$ The algebraic sum of the perpendicular drawn from $\left(2,0\right),\left(0,2\right)$ and $\left(1,1\right)$ on the line is zero, then the coordinates of the $P$ are

• A. $\left(1,-1\right)$
• B. $\left(1,1\right)$
• C. $\left(2,1\right)$
• D. $\left(2,2\right)$

Answer 1

The correct option is B

$\left(1,1\right)$

Explanation for correct option:

Step-1: Express the given data.

Let the variable line passing through point $P$ is $ax+by+c=0..........\left(i\right)$

Given, the sum of distance from the points $\left(2,0\right),\left(0,2\right)$ and $\left(1,1\right)$ to line $ax+by+c=0$ is zero.

Step-2: Apply formula for finding distance from a point $\left(x_1,y_1\right)$ to line $ax+by+c=0$ is $\frac{\left|a{x}_1+b{y}_1+c\right|}{\sqrt{a^2+b^2}}$.

Distance from point $\left(2,0\right)$ to line $ax+by+c=0$ is $\frac{\left|2a+0b+c\right|}{\sqrt{a^2+b^2}}......\left(ii\right)$

Distance from point $\left(0,2\right)$ to line $ax+by+c=0$ is $\frac{\left|0a+2b+c\right|}{\sqrt{a^2+b^2}}......\left(iii\right)$

Distance from point $\left(1,1\right)$ to line $ax+by+c=0$ is $\frac{\left|1a+1b+c\right|}{\sqrt{a^2+b^2}}.....\left(iv\right)$

Step-3: According to question $\left(ii\right)+\left(iii\right)+\left(iv\right)=0$

$\Rightarrow$$\frac{\left|2a+0b+c\right|}{\sqrt{a^2+b^2}}+\frac{\left|0a+2b+c\right|}{\sqrt{a^2+b^2}}+\frac{\left|1a+1b+c\right|}{\sqrt{a^2+b^2}}=0$

$\Rightarrow$ $a+b+c=0........\left(v\right)$

Compare equation $\left(i\right)\&\left(v\right)$

$\Rightarrow$ $a=1\&b=1$

Therefore, the required point is $\left(1,1\right)$

Hence, option $\left(B\right)$ is correct.