Question

The number of points on the line $x+y=4$, which are a unit distance apart from the line $2x+2y=5$, is

• A. $0$
• B. $1$
• C. $2$
• D. $\infty$

Answer 1

The correct option is A

$0$

Explanation for the correct option:

Step 1: Find the slope of both the lines.

Equations of two lines are given.

$$\begin{gathered} x+y=4...1 \\ \Rightarrow 2x+2y=5...2 \end{gathered}$$

Find the slope of both the lines.

Equation $1$ can be rewritten as:

$y=-x+4$

So, the slope of equation $1$ is $-1$.

Equation $2$ can be rewritten as:

$y=-x+\frac{5}{2}$

So, the slope of equation $2$ is $-1$.

Step 2: Find the required number of points.

Since, the slope of both the lines is equal.

Therefore, both the lines are parallel.

So, the distance $d$ between both the lines is given by:

$$\begin{gathered} d=\left|\frac{4-{\displaystyle \frac{5}{2}}}{\sqrt{1^2+1^2}}\right| \\ \Rightarrow d=\left|\frac{{\displaystyle \frac{3}{2}}}{\sqrt{2}}\right| \\ \Rightarrow d=\frac{3}{2\sqrt{2}} \\ \Rightarrow d=\frac{3\sqrt{2}}{4}>1 \end{gathered}$$

Therefore, there is no point on the line $x+y=4$, which is a unit distance apart from the line $2x+2y=5$.

Hence, option $A$ is the correct option.