The point on the line -1x+1y + 5 = 0 which is closest to the point(-2,2) has x coordinate ______ and?

Question

y coordinate _________.

Answer 1

-1x+1y + 5 = 0 --------> y= x-5
So min (-2,2) ----- (x,y)
min (x+2)^2+(x-5-2)^2
min (x+2)^2+(x-7)^2
min x^2+4x+4 + x^2-14x+49
min 2x^2-10x+53
Take derivative and set to 0
4x-10=0
x=10
x=2.5

So y=-2.5

Answer 2

Given line:
-x + y + 5 = 0
-x + y = -5
x - y = 5

The point on the above line closest to the point (-2, 2) will be on the line thru (-2, 2) that is perpendicular to the given line. So the slope will be the negative reciprocal of the slope of the given line. Swap the coefficients of x and y and change the sign of one of them. Let's chosse the y coefficient.

(x + 2) + (y - 2) = 0
x + y = 0

Now find the intersection of the two lines. It will be the closest point.

x - y = 5
x + y = 0

Add the two equations.

2x = 5
x = 5/2

Plug the value for x into the second equation and solve for y.

x + y = 0
5/2 + y = 0
y = -5/2

The closest point is (x, y) = (5/2, -5/2).