Find the point on the line y = − 3 x + 2 closest to the point ( 6 , 6 )

Question

Find the point on the line y = − 3 x + 2 closest to the point ( 6 , 6 )

The function giving the distance between the point and the line is

s=  (Enter a function of x) 

The point on the line y=−3x+2 closest to (6,6) is   (Enter the coordinates of the point). Be sure to include commas and parentheses as required.

Answer 1

Here's a calculus method:

s = √[(x - 6)² + (-3x + 2 - 6)²]

s = √[(x² - 12x + 36) + (9x² + 24x + 16)]

s = √(10x² + 12x + 52)

s^' = (20x + 12) / 2√(10x² + 12x + 52) = 0

x = - 3/5 ; y = -3 (- 3/5) + 2 = 19 / 5 (-3/5 , 19/5)

As a quick check, we can make sure that this point and the point (6 , 6) lie on a line with slope 1/3. (Since the perpendicular distance from the point to the line will be the shortest distance to the line.)

30/5 - 19/5 / 30/5 - -3/5 = 11/33 = 1/3

Answer 2

Let this point is (x, y). then y = - 3x + 5.

Then d(x) = (x - 6)² + (- 3x + 5 - 6)² = x² - 12x + 36 + 9x² + 6x + 1 = 10x² - 6x + 37;

d'(x) = 20x - 6 = 0; x = 0.3, y = 4.1

d''(x) = 20 > 0. So, we have minimum. Point is (0.3, 4.1)