Find the point on the line

Question

Find the point on the line 5x+3y+1=0 which is closest to the point (–1,3).

 

 

x= ?

y=?

 

Answer 1

Slope of the line 3y+5x+1=0

is.  -5/3

Slope of line perpendicular to this line is 3/5

Equation of line passing through (-1,3) and Wirth a slope 3/5

 

is

 

y-3=(3/5)(x+1)

 

5y-3x=18

Point of intersection of the two lines

5y-3x=18.    and

5x+3y=-1

 

x=-177/102

y=87/34

 

For a check of answer----

This Point should  satisfy the equation

5x+3y=-1

Answer 2

First rewrite this equation in the for y = mx + b

5x + 3y + 1 = 0 ⇒ y = (-5/3)x - 1/3

 

The point on this line closest the point (-1,3) should be where a perpendicular line cross this point and the line of our equation.

 

The equation of the perpendicular line has a slope equal to -1/m

y = (3/5)x + b

 

Plug in the point to find the y intersect

3 = (3/5)(-1) + b ⇒ b = 18/5

So y = (3/5)x + 18/5

 

Now we have to find where these two lines intersect. We two equations and two unknown. 

y = (-5/3)x - 1/3

y = (3/5)x + 18/5

So, (-5/3)x - 1/3 = (3/5)x + 18/5 ⇒ (-25/15)x - 5/15 = (9/15)x + 54/15 ⇒ (-25/15 - 9/15)x = 5/15 + 54/15 ⇒ 

(-34/15)x = 59/15 ⇒ x = -59/34

y = (3/5)(-59/34) + 18/5 = -177/170 + 612/170 = 435/170 = 87/34

 

The closest point is (-59/34,87/34)

Answer 3

5x + 3y + 1 = 0  is a linear equation, we can put this in slope-intercept form.

 

3y = -5x - 1

 

y = (-5/3)x - (1/3)

 

 

Let use the slope and y-intercept of this linear function to find points on this line that is closest to the point (-1, 3).

 

x = 0       y = -1/3

 

x = -1      y = 4/3

 

x = -2      y = 3

 

x = -3      y = 14/3

 

 

If we analyze these points, we will find that the closest point is

 

x = -2

y = 3

 

 

You can also find the distance between the point (-1, 3) and each of the points we listed on the given line.  Go with the point that has the shortest distance.