# Thread: Straight Line Geometry Q

1. ## Straight Line Geometry Q

The straight line L passes through the points A(6,4) and B(2,10)

1) Find an equation of L expressing your answer in the form ax+by+c=0

The point (-2,k) lies on the line L

2) Find the value of k

The line M has an equation y-x+12=0. Lines M and L intersect at point P.

3) Find the coordinates of point P.

2. Originally Posted by x-disturbed-x
The straight line L passes through the points A(6,4) and B(2,10)

1) Find an equation of L expressing your answer in the form ax+by+c=0

The point (-2,k) lies on the line L

2) Find the value of k

The line M has an equation y-x+12=0. Lines M and L intersect at point P.

3) Find the coordinates of point P.
1) Two known points on the line.
First use the point-slope form (y-y1) = m(x-x1), then transform to the general form Ax +By +C = 0

m = (y2 -y1)/(x2 -x1)
m = (10 -4)/(2 -6) = 6/(-4) = -3/2
y -4 = (-3/2)(x -6)
Clear the fraction, multiply both sides by 2,
2y -8 = -3x +18
3x +2y -8 -18 = 0
3x +2y -26 = 0 ----------answer.

2) If point (-2,k) is on line L, then,
3(-2) +2(k) -26 = 0
-6 +2k -26 = 0
2k = 6 +26 = 32
k = 32/2 = 16 --------answer.

3.)
Line M is y -x +12 = 0
Line L is 3x +2y -26 = 0
If they intersect at point P, then their coordinates are the same at point P.
Or, the y of line M equals the y of line L at point P.

y -x +12 = 0
y = x -12 -------y of line M anywhere.

Substitute that into the equation of line L,
3x +2(x -12) -26 = 0
3x +2x -24 -26 = 0
5x -50 = 0
5x = 50
x = 50/5 = 10 -----the x-coordinate of point P.
Substitute that into the y of line M,
y = 10 -12
y = -2 ----the y-coordinate of point P

Therefore, point P is (10,-2) --------answer.

Check that on line M,
y -x +12 = 0
-2 -10 +12 =? 0
0 =? 0
Yes, so, OK.

On line L,
3x +2y -26 = 0
3(10) +2(-2) -26 =? 0
30 -4 -26 =? 0
0 =? 0
Yes, so, OK also.