<PRE><B>Question 65750</B>
&lt;pre&gt;&lt;font size = 4&gt;&lt;b&gt;determine the ratio in which the line joining (0,7) and (-2,1)is 
divided by the line 2x+y-4=0

&lt;i&gt;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.&lt;/i&gt;


Here&#39;s a graph of the line segment joining (0,7) and (-2,1) in red

{{{ graph( 133.3, 300, -3, 1, -1, 8, ((sqrt(-x))/(sqrt(-x)))*((sqrt(x+2))/(sqrt(x+2)))*(3x+7)  ) }}}

Now we&#39;ll add the graph of 2x + y - 4 = 0 in green

{{{ graph( 133.3, 300, -3, 1, -1, 8, ((sqrt(-x))/(sqrt(-x)))*((sqrt(x+2))/(sqrt(x+2)))*(3x+7),4-2x  ) }}}

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:

     y&lt;sub&gt;2&lt;/sub&gt; - y&lt;sub&gt;1&lt;/sub&gt;
m = ЧЧЧЧЧЧЧЧЧ
     x&lt;sub&gt;2&lt;/sub&gt; - x&lt;sub&gt;1&lt;/sub&gt;

where (x&lt;sub&gt;1&lt;/sub&gt;, y&lt;sub&gt;1&lt;/sub&gt;) = (0,7) and (x&lt;sub&gt;2&lt;/sub&gt;, y&lt;sub&gt;2&lt;/sub&gt;) = (-2,1)

     (1) - (7)     -6     
m = ЧЧЧЧЧЧЧЧЧЧЧ = ЧЧЧЧ = 3
     (-2) - (0)    -2      

Now substitute in the point slope formula:

y - y&lt;sub&gt;1&lt;/sub&gt; = m(x - x&lt;sub&gt;1&lt;/sub&gt;)

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 = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;(x&lt;sub&gt;2&lt;/sub&gt; - x&lt;sub&gt;1&lt;/sub&gt;)&lt;sup&gt;2&lt;/sup&gt;+(y&lt;sub&gt;2&lt;/sub&gt; - y&lt;sub&gt;1&lt;/sub&gt;)&lt;sup&gt;2&lt;/sup&gt;

 to find the distance from (0,7) to (-.6, 5.2)

        _____________________
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;(-.6 - 0)&lt;sup&gt;2&lt;/sup&gt;+(5.2 - 7)&lt;sup&gt;2&lt;/sup&gt;
        __________
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;.36 + 3.24
        ___
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;3.6

Also use it to find the distance from (-.6, 5.2) to (-2,1)
        _________________________
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;(-2 - (-.6) )&lt;sup&gt;2&lt;/sup&gt;+(1 - 5.2)&lt;sup&gt;2&lt;/sup&gt;
        ____________________
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;(-2 + .6)&lt;sup&gt;2&lt;/sup&gt; + (-4.2)&lt;sup&gt;2&lt;/sup&gt;
        _______________
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;(-1.4)&lt;sup&gt;2&lt;/sup&gt; + 17.64
        ____________ 
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;1.96 + 17.64
        ____
   d = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;19.6
                                    
4. The ratio of the two distances &quot;longer to shorter&quot; is
    ____     ___
i  &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;19.6 to &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;3.6, which can be
   expressed as a fraction
     ________     ______    ___  __
    &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;19.6/3.6  = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;196/36 = &lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;196/&lt;font face = &quot;symbol&quot;&gt;╓&lt;/font&gt;36 = 14/6 = 7/3 or 7 to 3 or 7:3.

The ratio &quot;shorter to longer&quot; is 3 to 7 or 3:7

Edwin&lt;/pre&gt;

 
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