A line segment is bisected by a line with the equation # -7 y + 3 x = 2 #. If one end of the line segment is at #( 2 , 4 )#, where is the other end?
Infinite solutions: any of the points in the line
First we should notice that the point (2,4) isn't in the line
But as the problem is stated it admits infinite solutions.
Calling AB the line segment with M the midpoint we have:
The orthogonal projection of A onto the line is also M when the segment is perpendicular bisected.
The slope of the line
So the slope of line perpendicular to the aforementioned is
And the equation of such a perpendicular line passing through A is
Combining the equations of the 2 lines
So when the segment AB is perpendicularly bisected by the line mentioned in the problem
But since it is not required that the bisection happens perpendicularly the set of possible points B lays in a line parallel to the bisecting line and passing through