Question

The coordinates of the foot of the perpendicular from $(0,0)$upon the line $x+y=2$ are

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

Answer 1

The correct option is C

$(1,1)$

Explanation for correct option:

Finding the coordinates:

Given $x+y=2...\left(i\right)$

Visualizing the given information.

Slope of the line

$$\begin{gathered} x+y=2 \\ \Rightarrow y=-x+2 \end{gathered}$$

Slope $=-1$

So, the slope of $AB$ will be $=1$

Now, $AB$ passes through the points $(0,0)$.

Therefore the equation of $AB$ will be

$$\begin{gathered} y-y_1=\text{slope}\left(x-x_1\right) \\ y-0=1\left(x-0\right) \\ \Rightarrow y=x \\ \Rightarrow y-x=0...\left(ii\right) \end{gathered}$$

Substituting equation $\left(ii\right)$ in $\left(i\right)$

$$\begin{gathered} \Rightarrow x+x=2 \\ \Rightarrow 2x=2 \\ \Rightarrow x=1 \end{gathered}$$

Therefore

$$\begin{gathered} y=x \\ y=1 \end{gathered}$$

The point $B$ is $(1,1)$

Hence, option (C) is the correct answer.