find the point on the line -4x+6y-2=0, which is closest to the point (-2,-3)

Question

Please answer the question.

 

Answer: (___ , ___)

Answer 1

1) determine the equation of the line perpendicular through (-2, -3)

2) determine the point of intersection with the given line.

3) this is an Algebra 2 question

Answer 2

We need a line going through (-2,-3) that is perpendicular to -4x+6y-2=0.  Let's first find the slope of that line, which will be the negative reciprocal of the given line.

 

-4x + 6y - 2 = 0

6y = 4x + 2

y = 2x/3 + 1/3

 

So the slope of the line will be -3/2.

 

y = -3x/2 + b

 

Plug in the point to find b.

 

-3 = -3(-2)/2 + b

-3 = 3 + b

b = -6

y = -3x/2 -6

 

Since the point will be common to both lines, set the equations equal to each other and solve for x.

 

2x/3 + 1/3 = -3x/2 - 6

4x + 2 = -9x - 36

13x = -38

x = -38/13

 

y = 2(-38/13)/3 + 1/3

3y = -76/13 + 1 = -63/13

y = -21/13

 

So the point is (-38/13, -21/13).  It seems weird, but it works.