Question 65750: determine the ratio in which the line joining (0,7) and (-2,1)is
divided by the line 2x+y-4=0
Found 2 solutions by Nate, Edwin McCravy:
Answer by Nate(3500)   (Show Source):
You can put this solution on YOUR website! Ratio of the slopes?
m = (7 - 1)/(0 + 2) = 6/2 = 3
y - 7 = 3(x + 0)
y = 3x + 7 is the equation just in case ....
second:
y = -2x + 4
slope: -2
Ratio of slopes: 3:-2 or 3/-2
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website! determine the ratio in which the line joining (0,7) and (-2,1)is
divided by the line 2x+y-4=0
Nate above took it that you wanted the ratio of the slopes. I think
you wanted the ratio in which the line divides the segment.
Here's a graph of the line segment joining (0,7) and (-2,1) in red
Now we'll add the graph of 2x + y - 4 = 0 in green
We want to know into what ratio the green line divides the red
line segment.
Plan:
1. Find the equation of the line joining (0,7) and (-2,1)
2. Solve the system of equations consisting of the results
of step 1 and line 2x+y-4=0 to find their point of
intersection.
3. Find the distances from the point found in step 3 to
(0,7) and (-2,1)
4. Find the ratio of these two distances.
1. Find the slope, m:
y2 - y1
m = —————————
x2 - x1
where (x1, y1) = (0,7) and (x2, y2) = (-2,1)
(1) - (7) -6
m = ——————————— = ———— = 3
(-2) - (0) -2
Now substitute in the point slope formula:
y - y1 = m(x - x1)
y - 7 = 3(x - 0)
y - 7 = 3x
y = 3x + 7
2. Solve the system:
2x + y - 4 = 0
y = 3x + 7
You can do this by substitution.
Point of intersection = (-3/5, 26/5) = (-.6, 5.2)
3. Use the distance formula
_____________________
d = Ö(x2 - x1)2+(y2 - y1)2
to find the distance from (0,7) to (-.6, 5.2)
_____________________
d = Ö(-.6 - 0)2+(5.2 - 7)2
__________
d = Ö.36 + 3.24
___
d = Ö3.6
Also use it to find the distance from (-.6, 5.2) to (-2,1)
_________________________
d = Ö(-2 - (-.6) )2+(1 - 5.2)2
____________________
d = Ö(-2 + .6)2 + (-4.2)2
_______________
d = Ö(-1.4)2 + 17.64
____________
d = Ö1.96 + 17.64
____
d = Ö19.6
4. The ratio of the two distances "longer to shorter" is
____ ___
i Ö19.6 to Ö3.6, which can be
expressed as a fraction
________ ______ ___ __
Ö19.6/3.6 = Ö196/36 = Ö196/Ö36 = 14/6 = 7/3 or 7 to 3 or 7:3.
The ratio "shorter to longer" is 3 to 7 or 3:7
Edwin
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