Problem: Given the line y=43​x+6 and a line L parallel to the given line and 4 units from it. A possible equation for L is:
Answer Choices:
A. y=43​x+1
B. y=43​x
C. y=43​x−32​
D. y=43​x−1
E. y=43​x+2
Solution:
There are two possibilities, L1​ and L2​. The methods for finding the equation of L2​ are similar to the methods for finding the equation of L1​ shown here: Since L1​ is parallel to the line y=43​x+6, its equation is y=43​x+b, where b, the y-intercept, is to be determined. Since △ABC∼△DAO,ADAB​=DOAC​, so that 10AB​=84​ and AB=5. Hence, b=1 and the equation of L1​ is y=43​x+1.
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or
Let d1​ be the distance from the origin to L1​ and let d2​ be the distance from the origin to the given line. Then d1​=54b​ and d2​=524​.∴d2​−d1​=524​−54b​=4, and b=1.
