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

Aganippe76 2022-07-01 Answered
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 ) .
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

eurgylchnj
Answered 2022-07-02 Author has 14 answers
Step 1
If 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 2
The formula for the distance between these two parallel lines is
d = | b 1 b 2 | m 2 + 1 .
Step 3
So if you know b 1 and have a particular d in mind, you can rearrange this formula to solve for the second line's parameter b 2 .
b 2 = b 1 ± d m 2 + 1
There are two solutions because the second line can be "above" or "below" the first line.
Step 4
In the special case that the lines are vertical, the slope m is infinite so this formula doesn't work. Instead, our lines are simply x = b 1 and x = b 1 ± d

We have step-by-step solutions for your answer!

Joel French
Answered 2022-07-03 Author has 10 answers
Step 1
Using 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 .

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions