Perpendicular line, y-intercept

Question

Lines l and m are perpendicular lines that intersect at the point (3a,a-1), where a is a constant. Line l is represented by the equation 6x - 4y = k, where k is a constant. Find the value of a if the y-intercept of line m is (0,k).

Answer 1

Using the Standard Form of a Line:    lines Ax + By = C  and  Bx - Ay = D  are perpendicular.

 

The problem gives Line l as:  6x-4y=k with point (3a,a-1), so,

                                   6(3a)-4(a-1)=k

                                      14a+4 = k             (later remember, k is a constant)

 

The parallel line m is:  -4x-6y = D with y-intercept (0,k) as well as point (3a,a-1), so,

                                             -6(k)=D   as well as   -4(3a)-6(a-1)=D

                                                                                     -12a-6a+6=D

                                                                                         -18a+6=D

       Making these equal:    D= -6k = -18a + 6

                                                         k = 3a - 1           (divide both sides by -6)

Now, values of constant k:

                                       14a+4 = k = 3a-1

                                         11a = -5                 (collect terms)

                                            a = -5/11         (divide both sides by 11)

 

                                        also,  k = 3a-1 = -26/11     (needed for checking)

                                        also, D=-6k=-6(-26/11)=156/11      (needed for checking)

Checking (very important):

      Do line l (6x-4y= -26/11) and line m (-4x-6y=156/11)

                share [intersect at] point (3a,a-1), that is (-15/11,-16/11)  ??

                      6(-15/11)-4(-16/11) = -26/11     and       -4(-15/11)-6(-16/11) = 156/11    ?

                         -90/11 + 64/11 = -26/11        and          60/11 + 96/11 = 156/11   ?

                               -26/11   = -26/11              and          156/11 = 156/11    ?    Yes !

Answer 2

6x - 4y = k

 

      -4y = -6x + k

 

         y = (3/2)x -(1/4)k   ← slope = 3/2

 

The "other" line has slope -2/3 and y-intercept (0, k)

 

        Equation of this line is y = (-2/3)x + k

 

(3a, a-1) is on both lines, so:  a - 1 = (9/2)a - (1/4)k

                                           and  a - 1 =  -2a + k

 

Simplifying, we get:  (7/2)a - (1/4)k = -1

                                         -3a + k         = -1

 

Multiply the first equation by 4 :

 

      14a - k = -4

      -3a + k = -1

 

 Add the equations:

 

      11a = -5

 

          a = -5/11