Find the point of the line

Question

Find the point on the line 5x+3y+3=0 which is closest to the point (-3,3).

 

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Answer 1

Another way to do this problem without calculus. Find the equation of the line that passes through (-3,3) and is perpendicular to the given line (the slope of the perpendicular must be 3/5. Once you have the two equations find the point of intersection (perhaps by the substitution method). https://www.desmos.com/calculator/w9jrr8ditn

Answer 2

You need to find a line perpendicular to the given line and that passes through (-3,3).  The intersection of those 2 lines is the point you seek.  The slope of the given line is -5/3.  The slope of the perpendicular line must be

 

m = -1/(-5/3) = 3/5

 

The perpendicular line is

 

y = (3/5)x + b

3 = (3/5)(-3) + b

b = 15/5 + 9/5 = 24/5

 

y = (3/5)x + 24/5 ⇒ 

 

-3x + 5y = 24

 

Now solve for x and y in the system below, which I am sure you know how to do:

 

 5x + 3y = -3

-3x + 5y = 24

 

You should get

 

x = -87/34

y = 111/34

Answer 3

First, we solve for y from the given equation.

 

3y = -5x - 3

 

  y = (-5/3)x - 1

 

 

The point on this line is (x , (-5/3)x - 1).  It must be closer to the point (-3, 3).  If we use the distance formula, and find the minimal distance using first derivative test, we can find our point.

 

d = √[(x + 3)² + ((-5/3)x - 1 - 3)²]

 

d = √[(x² + 6x + 9) + ((-5/3)x - 4)²]

 

d = √[(x² + 6x + 9) + ((25/9)x² + (40/3)x + 16)]

 

d = √[(34/9)x² + (58/3)x + 25]

 

 

Now we set the derivative of this distance equal to zero.

 

(98/9)x² + (58/3)x = 0

 

 

Multiply both sides of the equation by 9.

 

98x² + 174x = 0

 

x(98x + 174) = 0

 

x = 0         or            98x + 174 = 0

                                98x = -174

                                   x = -7.286

 

Plug in these values of x into the original equation.

 

(0, -1)            and             (-7.286, 11.143)

 

 

(0, -1) is much closer.