Question

I am seeking help with this problem Please...Thank you!

Find the point on the line y=x+3 in the xy-plane that is closest to the point (3, 2)

Answer 1

Let P(x,y) be that point.
Would you agree, that is P is the closest point, then the line from P to (3,2) would be perpendicular to the given line?
slope of line = 1
so slope of perpendicular = - 1
slope of perp. = (y-2)/(x-3)
so (y-2)/(x-3) = -1
y - 2 = -x + 3
but y = x+3
x + 3 - 2 = -x + 3
2x = 4
x = 2
then y = 2 + 3 = 5

the closest point is (2,5)