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Question
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0 is ______.
Options
1 : 2
3 : 7
2 : 3
2 : 5
Solution
The ratio in which the line 3x + 4y + 2 = 0 divides the distance between the lines 3x + 4y + 5 = 0 and 3x + 4y – 5 = 0 is 3 : 7.
Explanation:
The given equations are
3x + 4y + 5 = 0 .......(i)
3x + 4y – 5 = 0 .....(ii)
And 3x + 4y + 2 = 0 ......(iii)
Clearly, equation (i), (ii) and (iii) are parallel to each other as the coefficients of x and y are same.
Distance between parallel lines (i) and (iii) we get
`|(5 - 2)/sqrt((3)^2 + (4)^2)| = 3/5` ......`[(because "Distance between two"),("parallel lines" = |("c"_1 - "c"_2)/sqrt("a"^2 + "b"^2)|)]`
Distance between parallel lines (ii) and (iii) we get
`|(-5 - 2)/sqrt((3)^2 + (4)^2)| = 7/5`
∴ Ratio between the distances = `3/5 : 7/5` = 3 : 7
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