Question

The line passing through the point of intersection of $x+y=2,x-y=0$and is parallel to $x+2y=5$is

• A. $x+2y=1$
• B. $x+2y=2$
• C. $x+2y=4$
• D. $x+2y=3$

Answer 1

The correct option is D

$x+2y=3$

Step 1: Given data

1. Let the line be$L$
2. $L$ is parallel to the line $x+2y=5$
3. The point on the line $L$ is the intersection of lines $x+y=2$and $x-y=0$

Step 2: Find the slope of the line $x+2y=5$

The slope of $L$ = Slope of the line $x+2y=5$

$m=-\frac{1}{2}$

Step 3: Find the point of intersection

Let the point be $(x_1,y_1)$

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

Putting the value in the second equation

$$\begin{gathered} \Rightarrow x-y=0 \\ \Rightarrow x-(2-x)=0 \\ \Rightarrow x=1 \\ \therefore x_1=1 \end{gathered}$$

Now, the value of $y$

$$\begin{gathered} \Rightarrow y=2-x \\ \Rightarrow y=2-1=1 \\ \therefore y_1=1 \end{gathered}$$

Step 4: Find the equation of the line

The equation of the line passing through $\left(x_1,y_1\right)$ and slope $m$ is

$\left(y-y_1\right)=m\left(x-x_1\right)$

$$\begin{gathered} \Rightarrow (y-1)=\frac{-1}{2}(x-1) \\ \Rightarrow 2y-2=-x+1 \\ \Rightarrow x+2y=3 \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{D}\right)$.