A(2,8), B(6,4) and C(-6,y) are collinear points find y?

3 Answers
Feb 21, 2018

y=16y=16

Explanation:

If a set of points are collinear the belong to the same straight line, whose generale equation is y=mx+qy=mx+q
If we apply the equation to the point A we have:
8=2m+q8=2m+q
If we apply the equation to the point B we have:
4=6m+q4=6m+q
If we put this two equation in a system we can find the equation of the straight line:

  1. Find mm in the first eq.
    m=(8-q)/2m=8q2
  2. Replace mm in the second eq. and find qq
    4=6(8-q)/2=>4=3(8-q)+q=>4=24-3q+q=>-20=-2q=>q=104=68q24=3(8q)+q4=243q+q20=2qq=10
  3. Replace qq in the first eq.
    m=(8-10)/2=-1m=8102=1
    Now we have the equation of the straight line:
    y=-x+10y=x+10
    If we replace C coordinates in the equation we have:
    y=6+10=>y=16y=6+10y=16
Feb 21, 2018

1616.

Explanation:

Prerequisite :

"The points "(x_1,y_1),(x_2,y_2) and (x_3,y_3)" are collinear"The points (x1,y1),(x2,y2)and(x3,y3) are collinear

hArr |(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|=0.

Therefore, in our Problem, |(2,8,1),(6,4,1),(-6,y,1)|=0,

rArr 2(4-y)-8{6-(-6)}+1{6y-(-24)}=0,

rArr 8-2y-96+6y+24=0,

rArr 4y=64,

rArr y=16, as Respected Lorenzo D. has already derived!.

Feb 21, 2018

P_C->(x_c,y_c)=(-6,+16)

Full details shown. With practice you will be able to do this calculation type with very few lines.

Explanation:

color(blue)("The meaning of 'collinear'")

Lets split it into two parts

color(brown)("co"->"together". Think about the word cooperate
color(white)("ddddddddddddd")So this is 'together and operate.'
color(white)("ddddddddddddd")So you are doing some operation (activity)
color(white)("ddddddddddddd")together

color(brown)("liniear".->color(white)("d") In a strait line.

color(brown)("collinear")-> co =together, linear =on a strait line.

color(brown)("So all the points are on a strait line")
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Answering the question")

color(purple)("Determine the gradient (slope)")

The gradient for part is the same as the gradient for all of it

Gradient (slope) ->("change in y")/("change in x")

Set point P_A->(x_a,y_a)=(2,8)
Set point P_B->(x_b,y_b)=(6,4)
Set point P_C->(x_c,y_c)=(-6,y_c)

The gradient ALWAYS reads left to right on the x-axis (for standard form)

So we read from P_A " to " P_B thus the we have:

Set gradient -> m="last "-" first"

color(white)("d")"gradient " -> m=color(white)("d")P_Bcolor(white)("d")-color(white)("d")P_A

color(white)("dddddddddddd")m=color(white)("d,")(y_b-y_a)/(x_b-x_a)

color(white)(dddddddddddddddddddd") (4-8)/(6-2) = -4/4=-1

Negative 1 means that the slope (gradient) is downward as you read left to right. For 1 across there is 1 down.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(purple)("Determine the value of "y)

Determined that m=-1 so by direct comparison

P_C-P_A =m = (y_c-y_a)/ (x_c-x_a) = -1

color(white)("dddddddddddd.d") (y_c-8)/ (-6-2) = -1

color(white)("dddddddddddddd.") (y_c-8)/ (-8) = -1

Multiply both sides by (-8)

color(white)("ddddddddddddddd.") y_c-8 = +8

Add 8 to both sides

color(white)("ddddddddddddddddd.")y_c color(white)("d")=+16

Tony BTony B