Find the point on the line 5x+4y−4=0 which is closest to the point (4,0).

Question

Find the point on the line 5x+4y−4=0which is closest to the point (4,0).

Answer 1

The easiest way is to use the method described by Raymond B in his response to the question. If you want to se the Calculus way, first make a sketch:

Then write an equation for the distance using the Pythagorean Theorem between the two points. HOWEVER, for the point (x, y) use the relationship that y = -(5/4)x + 1 so the point becomes (x, -(5/4)x+1)

Once the distance function is written, take the derivative and set it equal to zero and solve.

Answer 2

5x+4y =4

y = 1 -5x/4

perpendicular line is

y = 4x/5 + b

plug in (4,0)

0 = 16/5 +b

b = -16/5

y = 4x/5 -16/5

find where it intersects the original line

-5x/4+1 = 4x/5 -16/5

4x/5 +5x/4 = 1+16/5

16x+25x = 20+64

41x = 84

x = 84/41

y = -(5/4)(84)/41 +1 =-105/41 +41/41 = -64/41

closet point on the line to (4,0) is

(84/41, -64/41)= about (2.05, -1.56) or (2.1,-1.6)

it may help to draw a rough sketch of the lines and points

the closest point on a line to another point not on the line is found by a perpendicular line connecting the two points