s_bilgrami89
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question 14 23(b) from the 3u book
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need help with interval question fitzpatrick (1 Viewer)
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SpiralFlex
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Show that the point )
lies on the line joining )
and )
and find the ratio in which is divides 
First we must find the equation of
We can always find the gradient,
Now, we use a infamous Year 8 formula,
)
)
Simplify!
We must test to see if the equation holds true at point P. Testing,)
+2=0)
Hence
lies on the interval
There is one thing we must do before attempting this question, actually, two things. We must first understand if it is an internal or external division. Then we must understand what we are actually doing when doing division of an interval.
Drawing a pretty diagram will help this cause!
From our pretty diagram, it is obviously an internal division! So no minuses involved. We can also see that obviously by similar triangles,
Similarly,
Ah, now if we simplify each expansion we get our division formula! But we do not want to do so because, we have what we need.
Just substitute our values inside! One of the equations! Let's pick first since first is the best and good for your ATAR.
Horaay! Therefore the ratio that divides 
is in,
You must understand the underlying concept of ratio division! No rote learning!
First we must find the equation of
We can always find the gradient,
Now, we use a infamous Year 8 formula,
Simplify!
We must test to see if the equation holds true at point P. Testing
Hence
There is one thing we must do before attempting this question, actually, two things. We must first understand if it is an internal or external division. Then we must understand what we are actually doing when doing division of an interval.
Drawing a pretty diagram will help this cause!
From our pretty diagram, it is obviously an internal division! So no minuses involved. We can also see that obviously by similar triangles,
Similarly,
Ah, now if we simplify each expansion we get our division formula! But we do not want to do so because, we have what we need.
Just substitute our values inside! One of the equations! Let's pick first since first is the best and good for your ATAR.
Horaay! Therefore the ratio that
You must understand the underlying concept of ratio division! No rote learning!
Last edited:
I love the professional diagrams Spi keeps producing. Wish I knew how to.
Another way to show P lies on AB
To say that point P lies on the line AB is equivalent to saying that points A, B and P are collinear.
gradient of AP = [0 - (-3)]/[-2-7] = -1/3
gradient of PB = [1-0]/[-5 - (-2)] = - 1/3
Since gradient of AP = gradient of PB, P lies on line AB.
Another way to show P lies on AB
To say that point P lies on the line AB is equivalent to saying that points A, B and P are collinear.
gradient of AP = [0 - (-3)]/[-2-7] = -1/3
gradient of PB = [1-0]/[-5 - (-2)] = - 1/3
Since gradient of AP = gradient of PB, P lies on line AB.
hey spiralflex, i'm not completely sure what you mean by the division formula not working...either i've misunderstood what you meant or there's something wrong with my working because i managed to get the answer using the formula.
letting the ratio that P divides the interval AB be k:d,
[7d-5k]/[k+d] = -2 from the x-values of A, B and P
7d-5k = -2k - 2d
9d=3k
.'. 3d=k --> d=k/3
doing the same thing with the y-values, you get the same result with d=k/3, but if you go back to the ratio and see that it is k:d, subbing the value of d into the ratio, you get the ratio k:k/3 which simplifies to be 3:1.
letting the ratio that P divides the interval AB be k:d,
[7d-5k]/[k+d] = -2 from the x-values of A, B and P
7d-5k = -2k - 2d
9d=3k
.'. 3d=k --> d=k/3
doing the same thing with the y-values, you get the same result with d=k/3, but if you go back to the ratio and see that it is k:d, subbing the value of d into the ratio, you get the ratio k:k/3 which simplifies to be 3:1.
SpiralFlex
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Sorry, I meant something completely different, I was trying to show where it comes from.