Question

The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the line 3x + 4y + 5 = 0 and 3x + 4y − 5 = 0 is
(a) 1: 2
(b) 3: 7
(c) 2: 3
(d) 2: 5

Answer 1

Here, in all equations the coefficient of x is same.
It means all the lines have same slope
So, all the lines are parallel.
Now, the distance between the line 3x + 4y + 2 = 0 and 3x + 4y + 5 = 0 is given by
$$\begin{gathered} \frac{\left|2-5\right|}{\sqrt{3^2+4^2}} \\ =\frac{3}{\sqrt{25}}=\frac{3}{5} \end{gathered}$$
Again, the distance between the line 3x + 4y + 2 = 0 and 3x + 4y − 5 = 0 is given by:

$$\begin{gathered} \frac{\left|2+5\right|}{\sqrt{3^2+4^2}} \\ =\frac{7}{25}=\frac{7}{5} \end{gathered}$$
Hence, the ratio is given by
$$\begin{gathered} \frac{3}{5}:\frac{7}{5} \\ =3:7 \end{gathered}$$


Hence, the correct answer is option (b).