Find the value of k if the line joining point D (8, k) and point E (3, -9) is perpendicular to the line 3x-2y=11

Question

Answers must be in fractions and/or radicals NOT rounded decimals 

Answer 1

The line 3x-2y=11 is in Standard Form (Ax+By=C) with slope (-A/B) or (3/2).  All perpendicular lines have negative reciprocal slope of (-2/3) and an equation of 2x+3y=C   [note: (-A/B) is the new slope).

 

Use point E(3,-9) to determine C:

       2(3)+3(-9) = C

           -21 = C

  The equation of the perpendicular line is:    2x+3y=-21

 

Point D(8,k) is on the line, so:

        2(8)+3(k) = -21

             3k = -37

               k = -37/3

Answer 2

The line perpendicular to 3x - 2y = 11 will have a slope of -2/3. Why? Because perpendicular lines have negative reciprocal slopes. The slope of 3x - 2y = 11 is 3/2. We can determine this by writing the equation of the line in y = mx + b form.

 

3x - 2y = 11

 

-2y = -3x + 11

 

y = (3/2) x - 11/2

 

slope = (y₂ - y₁)/(x₂ - x₁)

 

We can insert points D and E into the above formula for slope to determine the value of k. It doesn't matter which one you select for D and which you select for E.

 

slope = (k - (-9))/(8 - 3) = -2/3

 

(k + 9)/5 = -2/3

 

3(k + 9) = -10

 

3k + 27 = -10

 

3k = -37

 

k = -37/3 = -12 1/3