Yes, parallel lines have different intercepts
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They go in different directions.
Lines with the same slope but with different y intercepts are parallel.
If the slopes are different the lines are neither - they intersect. They are parallel or coincident if the slopes are the same. Then, if the y-intercepts are the same they are coincident while if the y-intercepts are different, they are parallel.
If the lines have the same slope but with different y intercepts then they are parallel
A cone does not have any parallel lines