The shortest distance between these two parallel lines is 2.5
Given equations:
3x + 4y + 10 = 0 ----- (1)
3x + 4y + 20 = 0 ---- (20)
To find:
the shortest distance between the two lines
The slope and y-intercept of the first equation is calculated as;
3x + 4y + 10 = 0
4y = -3x - 10
[tex]y = \frac{-3x}{4} - \frac{10}{4} \\\\y = \frac{-3x}{4} - \frac{5}{2}[/tex]
The slope = -³/₄ and the y-intercept = - ⁵/₂ = - 2.5
The slope and y-intercept of the second equation is calculated as;
3x + 4y + 20 = 0
4y = -3x - 20
[tex]y = \frac{-3x}{4} - \frac{20}{4} \\\\y = \frac{-3x}{4} - 5[/tex]
The slope = -³/₄ and the y-intercept = - 5
The two equations have the same slope = -³/₄
This shows that they are parallel with different y-intercepts
The shortest distance between the two lines is at their y-intercepts
The shortest distance between these two lines is calculated as;
[tex]distance = \sqrt{(-5 - (-2.5) )^2} \\\\distance = \sqrt{(-5+ 2.5)^2} \\\\distance = \sqrt{(-2.5)^2 } \\\\distance = \sqrt{6.25} \\\\distance = 2.5[/tex]
Thus, the shortest distance between these two parallel lines is 2.5
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