In what ratio does the point 24
over 11 𝑦 divide the line segment joining the points 𝑃: two, negative two and 𝑄:
three, seven? Also, find the value of 𝑦.
In these sorts of questions, it’s
always good to just sketch things out to get a rough idea of what’s going on. In this case, we’ve got point 𝑃
down here at two, negative two and 𝑄 up here at three, seven. And the other point let’s call that
point 𝑅. The other point will be between 𝑃
and 𝑄 because its 𝑥-coordinate, 24 over 11, is bigger than two but less than
Now from this sketch, we don’t
actually know whether this is going to be below the 𝑥-axis or above the
𝑥-axis. But we’ll worry about that
later. So in the question, we’re looking
for the ratio 𝑃𝑅 to 𝑅𝑄. Now 𝑃𝑅𝑄 is a straight line, so
the 𝑦-coordinates change at a constant rate or the line has a constant slope or
Now we were given the
𝑥-coordinates for all three points. So if we find the ratio of the
differences in the 𝑥-coordinates from 𝑃 to 𝑅 to 𝑅 to 𝑄, that’ll be the same as
the ratios of the differences in the 𝑦-coordinates and, what we’re looking for, the
ratio of the lengths of the line segments 𝑃𝑅 to 𝑅𝑄.
So looking at the 𝑥-coordinates of
𝑃 and 𝑅, we’re going from an 𝑥-coordinate of two up to an 𝑥-coordinate of 24
over 11. So to work out the difference in
those 𝑥-coordinates, we simply subtract the first from the second. And subtraction of fractions is
much easier when they’ve got a common denominator.
So instead of writing two, I’m
gonna write 22 over 11, because 22 over 11 is the same as two. Then 24 over 11 minus 22 over 11 is
just two over 11. And the 𝑥-coordinates of points 𝑅
and 𝑄 are 24 over 11 and three, so the difference in those is three minus 24 over
11. And again, to get the common
denominator, we can reexpress three as 33 over 11. Then we’ve got 33 over 11 minus 24
over 11, which is nine over 11. So the ratio 𝑃𝑅 to 𝑅𝑄 is two
elevenths to nine elevenths.
But we wouldn’t normally leave
those in fraction form. We can multiply both sides of that
ratio by 11. And the 11s will cancel, enabling
us to express that ratio, 𝑃𝑅 to 𝑅𝑄, in its simplest form as the lowest integer
values that you can have for that ratio; that’s two to nine.
Now we need to find the value of
𝑦. And to do that, we’re gonna find
the equation of the line that runs through points 𝑃 and 𝑄. Now the general form of an equation
of the straight line is 𝑦 is equal to 𝑚𝑥 plus 𝑐, where 𝑚 is the gradient or
slope of that line and 𝑐 is the 𝑦-intercept, where it cuts the 𝑦-axis.
Now we can work out the slope or
the gradient by taking the difference in the 𝑦-coordinates and dividing that by the
difference in the 𝑥-coordinates. So if we call the first coordinate
pair 𝑥 one, 𝑦 one and the second coordinate pair 𝑥 two, 𝑦 two, 𝑚 is defined as
being 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. So if we just substitute those
values in for 𝑥 one, 𝑥 two, 𝑦 one, and 𝑦 two, we get seven minus negative two
over three minus two. And seven take away negative two is
nine, and three take away two is one. So the slope is nine over one, or
just nine. So we now know that the equation of
that line is gonna be 𝑦 is equal to nine 𝑥 plus wherever it cuts the 𝑦-axis.
But we do know some points that are
on the line. So we know that, for example, when
𝑥 is equal to three, then 𝑦 is equal to seven. So we can substitute those values
in for 𝑥 and 𝑦 and work out the value of 𝑐. That means that seven is equal to
nine times three plus 𝑐. Well, nine times three is 27. And if I now subtract 27 from both
sides of that equation, I’ll find that negative 20 is equal to 𝑐, or 𝑐 is equal to
negative 20. And I can plug that in to my
original 𝑦 equals 𝑚𝑥 plus 𝑐 equation. So the equation of the line that
goes through these two points is 𝑦 is equal to nine 𝑥 minus 20.
All I now need to do is substitute
in the 𝑥-coordinate, 24 over 11, and I can find out the corresponding
𝑦-coordinate. So 𝑦 is equal to nine times the
𝑥-coordinate, 24 over 11, minus 20. Well, nine times 24 over 11 is the
same as nine times 20 over 11 plus nine times four over 11. And nine times 20 is 180, and nine
times four is 36. So that first term becomes 180 plus
36 over 11, which is 216 over 11.
Now rather than just subtracting
20, let’s turn that into a fraction that has a denominator of 11 as well. So we need to multiply 20 by
11. And that means we can express 20 as
220 over 11. Now we’ve got 216 over 11 minus 220
over 11. And that tells us that the value of
𝑦 is negative four over 11.