Find the point on the line −2x+3y−1=0 which is closest to the point (4,0).

Answer 1

Alex, there is the easy way to find the answer (use Algebra 1) and the hard way (use Calculus) and it looks like you have been asked to do this the hard way.

Easy way: find the point slope form of the perpendicular to the line -2x + 3y + -1 = 0 which passes through the point (4,0) and then find the intersection of these two lines (because the perpendicular is the shortest distance).

Hard way: use the distance formula to create a distance function, take the derivative to find the local minimum, check endpoints to confirm that the local minimum is the global minimum, and you have your answer.

Hope this helps, but please let me know--Andrew

Answer 2

The point on the given line that is closet to the given point is a point on the perpendicular to the given line.

Determine slope of given line

Make negative reciprocal of slope given line. This is slope of perpendicular

Determine equaion of perpendicular using slope of perpendicular and given point

Determine point of intersection of given line and perpendicular.