caculate the distance from each given point to the given line point : (-6,-2) Line : f(x)=2x-5

Question

step by step and show your work

Answer 1

The distance from a line to a point not on the line is the shortest (perpendicular) distance from the point to the line.  

 

y = 2x-5 has slope 2.

 

Any line perpendicular to y = 2x-5 has slope -1/2.

 

We are looking for the point, P, on the line such that the slope of the line through P and (-6,-2) is equal to -1/2.

 

Since P lies on the line, P = (x, 2x-5).

 

So, (2x-5-(-2))/(x-(-6)) = -1/2

 

             (2x-3)/(x+6) = -1/2

 

                    2(2x-3) = -1(x+6)

 

                       4x-6 = -x-6

 

                           5x = 0

 

                            x = 0   So, P = (0,-5)

 

The distance, D,  that we are looking for is the distance between (-6,-2) and (0,5).

 

D = √[(0-(-6))² + (-5-(-2))²] = √45 = 3√5 ≈ 6.7

 

  

Answer 2

You are going to use the distance formula:


d = √[(x₁ - x₂)² + (y₁ - y₂)²]
 
where the first point is the given point and the other point is any arbitrary point on the line f(x).  d is the distance.
 

d = √[(-6 - x)² + (-2 - (2x - 5))²]

 

d = √[(-6 - x)(-6 - x) + (3 - 2x)(3 - 2x)]

 

d = √[(36 - 12x + x²) + (9 - 12x + 4x²)]

 

d = √(45 - 24x + 5x²)