Coordinate Geometry-Finding ratio of a point splitting a line

Coordinate Geometry

Question: In what ratio is the line joining the points (4,2) and (3,-5) divided by the x-axis? Also, find the coordinates of the point of intersection.

Solution: Let the required ratio be(K:1) and the point on the x-axis be (x,0).
Taking (4,2)=(x1:y1) and (3,-5)=(x2:y2)
Since, y=(ky1+y1)/(k+1)
The ratio of m:n is 2:5 and the value of x is 26/7
(I am sorry that i am unable to use subscript in this web page)


My question is y is the solution using K:1 as the ratio. Secondly why is the solution starting with solving the value of Y (ordinate) and not X.

Kindly help.
ExcellearnerAsked:
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d-glitchConnect With a Mentor Commented:
You diagram is okay, but not quite to scale.  The rise is 7, the run is 1, so it should be closer to vertical.

They do not mention K:1 in the problem, and they did not really use it in the solution.  I think it is intended as a hint to use ratios.  Or maybe because the run of the line segment is 1.

Back to the x-axis.
  • Q1.   What is the equation of the x-axis?  
           A2.    y = 0     (Nothing about x)

  • Q2.   What does the x-axis do?
           A2.   It divides the real plane into three regions:
                   y = 0  on the axis
                   y > 0  above the axis
                   y < 0  below the axis     (Once again, nothing about x)

The term x-axis refers to a particular line.  Make a good drawing and don't worry about the terminology.
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d-glitchCommented:
You need to make a drawing to visualize the problem.
And you need to remember that the equation for the x-axis is y=0.
So there are two units of the line above the x-axis and five below.

Complete the big right triangle [with sidebar x=4 and y=-5] to calculate the ratios.
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aburrConnect With a Mentor Commented:
d-glitch's suggestion to draw a diagram is VERY good. Once you do so, almost no calculation is needed.
note the y ratio is 2/5. that means that the x ratio is 2/5.
so the x intercept is 3 + the ratio of the one x unit to get to 4. Your diagram will tell you how to do it.
( 3 + 5/7 = the 26/7 value of x which you got.
A good look at your diagram will let you fill in the steps to get your answer. If it is not self-evident, ask.
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as far as your question "Secondly why is the solution starting with solving the value of Y (ordinate) and not X."
I have not the faintest idea.
Diagrams often make things easy.
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ExcellearnerAuthor Commented:
d-glitch and aburr, thank you both for your comments.

The attached includes the diagram and the workings.

For now i will not argue with why Y has been chosen instead of X. But can you help me understand why K:1 (that is one in alpha and the other in numeric) has been used instead of m:n ratio.

Thank you

*I did not understand this statement: so the x intercept is 3 + the ratio of the one x unit to get to 4. Your diagram will tell you how to do it.
( 3 + 5/7 = the 26/7 value of x which you got
IMG_4007-coordinate-g.JPG
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