Question

The distance between the lines $3x+4y=9$ and $6x+8y=15$ is:

• A. $\frac{3}{2}$
• B. $\frac{3}{10}$
• C. $6$
• D. None of these

Answer 1

The correct option is B

$\frac{3}{10}$

Explanation for the correct option:

Step 1: Finding the slope of the given two lines

The given lines are: $3x+4y=9$ and $6x+8y=15$

Slope of $3x+4y=9$

$$\begin{gathered} y=\frac{1}{4}\left(9-3x\right)\left[y=mx+c\right] \\ =-\frac{3}{4} \end{gathered}$$

Slope of $6x+8y=15$

$$\begin{gathered} y=\frac{1}{8}\left(15-6x\right)\left[y=mx+c\right] \\ =-\frac{3}{4} \end{gathered}$$

We can observe that the slope of the two lines are equal.

Hence, the two lines are parallel to each other.

Step 2: Finding the distance between two lines

The general form of equation is $ax+by=c$

Comparing $3x+4y=9$ and $3x+4y=\frac{15}{2}$ with the general equation, we get

$$\begin{gathered} a=3 \\ b=4 \end{gathered}$$

$$\begin{gathered} c_1=9 \\ {c}_2=\frac{15}{2} \end{gathered}$$

Therefore, the distance between parallel lines $3x+4y=9$ and $3x+4y=\frac{15}{2}$ is given by

$=\frac{\left|\left(c_1-c_2\right)\right|}{\sqrt{a^2+b^2}}\dots \dots \left(1\right)$

By substituting the values in equation $\left(1\right)$ we get,

$$\begin{gathered} =\frac{\left|\left(9-{\displaystyle \frac{15}{2}}\right)\right|}{\sqrt{3^2+4^2}} \\ =\frac{3}{10} \end{gathered}$$

Hence, Option (B) is the correct answer.