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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
Options
1: 2
3: 7
2: 3
2: 5
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Solution
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
\[\frac{\left| 2 - 5 \right|}{\sqrt{3^2 + 4^2}}\]
\[ = \frac{3}{\sqrt{25}} = \frac{3}{5}\]
Again, the distance between the line 3x + 4y + 2 = 0 and 3x + 4y − 5 = 0 is given by:
\[\frac{\left| 2 + 5 \right|}{\sqrt{3^2 + 4^2}}\]
\[ = \frac{7}{25} = \frac{7}{5}\]
Hence, the ratio is given by
\[\frac{3}{5} : \frac{7}{5}\]
\[ = 3 : 7\]
Hence, the correct answer is option (b).
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