Two parallel lines with varying slopes and constant distance ( say 4 units) between them.Let D be a line with equation y = a x + b, with a a varying number.So D has a changing slope and is rotating about point P = ( 0 , b ) .

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Two parallel lines with varying slopes and constant distance ( say 4 units) between them.Let D be a line with equation   y  =  a  x  +  b, with a a varying number.So D has a changing slope and is rotating about point   P  =  (  0  ,  b  ) .

eurgylchnj · Accepted Answer

Step 1If two lines are parallel, they have the form   y  =  m  x  +      b    1    and   y  =  m  x  +      b    2   . That is, their slopes are the same and their intercepts are different.Step 2The formula for the distance between these two parallel lines is  d  =                    |                    b        1            −              b        2                    |                            m        2            +      1        .Step 3So if you know       b    1   and have a particular d in mind, you can rearrange this formula to solve for the second line&#039;s parameter       b    2   .      b    2    =      b    1    ±  d            m      2        +    1  There are two solutions because the second line can be &quot;above&quot; or &quot;below&quot; the first line.Step 4In the special case that the lines are vertical, the slope m is infinite so this formula doesn&#039;t work. Instead, our lines are simply   x  =      b    1   and   x  =      b    1    ±  d

Joel French · Accepted Answer

Step 1Using vector geometry, 4-or, more generally, d-equals the scalar rejection of                     P        Q            →       from the direction vector v of D. So,                    d                            =                  |                                                    P                Q                            →                                ×                                                    v                            ^                                |                                                  =                  1                                    a              2                        +            1                                    |                      (                                                            0                                                                                                  ±                    P                    Q                                                                                                0                                                      )                    ×                      (                                                            1                                                                              a                                                                              0                                                      )                    |                                                  =                  1                                    a              2                        +            1                                    |                      (                                                            0                                                                              0                                                                                                  ∓                    P                    Q                                                                        )                    |                                                  =                              P            Q                                              a              2                        +            1                                              P        Q                            =        d                              a            2                    +          1                .            Thus, the required equation is                    y                            =        a        x        +        b        ±        P        Q                            y                            =        a        x        +        b        ±        d                              a            2                    +          1                .

Two parallel lines with varying slopes and constant distance ( say 4 units) between them. Let D be

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