How many integer points lie between points A and В on the line segment

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Hope it helps.

Between the points (5,7) and (10,-3) I got the slope as -2. I then kept increasing x by 1 and accordingly reduced y by -2. Got 4 integer points until (10,-3)

Hi

Can someone please correct my thought process here. What i thought was to find the distance between the two and see the integers points on that distance.
Distance came out to be 11.18.Considering integer coordinates we should have 11 integer points?
I know i am wrong somewhere, but need help to mend my thought process.

Is there any other method to solve this?

Slope matters. For example, the distance between (0, 0) and (1, 10) is \(\sqrt{101} \approx 10\) but 0 integer points lie between them. On the other hand the distance between (0, 0) and (0, \(\sqrt{101}\)) is also \(\sqrt{101} \approx 10\) and 10 integer points lie between them.

Hope it's clear.

Hi Bunuel,
If the question states that "integer points lie between points A and В", it means that A and B are excluded?
I mistakenly chose 6 points given the context did not state clearly inclusive or exclusive.

Better question would make it clearer but if nothing is mentioned, between usually means not inclusive.

I have calculated the slope which comes out to be -2 and post this i am not able to reach or figure out the next step?

Can anyone please help.

Take the general equation of a line; \(y=mx+c\)

Now substitute the two points A and B,

A(5,7) :\(7=5m+c\)...................(1)
B(10,-3) :\(-3=10m+c\)...................(2)

Subtracting (1) and (2)

\(10=-5m\)
\(m=-2\)..................(3)

Substituting (3) in (2), we get c=17

Now the equation of the line is; \(y=-2x+17\)
Notice that the boundary of x is given by the x co-ordinates of points A & B. Hence 5<=x<=10

Since the question asks for the number of points between A & B. (keyword = between), there are 4 integer values of x.
x=6,7,8 and 9. We can safely say that for all these values of x, y will be an integer and hence the answer is 4 points or A

Let’s determine the equation of the line that connects the two given points. We can start with the slope.

slope = change in y/change in x

slope = (-3 - 7)/(10 - 5) = -10/5 = -2

So, we have:

y = -2x + b

Let’s substitute 5 and 7 for x and y, respectively, to calculate b.

7 = -2(5) + b

7 = -10 + b

17 = b

So, we have:

y = -2x + 17

We see that whenever x is an integer in this equation, y will also be an integer, and therefore (x, y) will be an integer point. Furthermore, when x is between 5 and 10, the corresponding points on the line will be between A and B. Thus, the points on the line whose x coordinates are 6, 7, 8, and 9 are the only integer points between A and B. There are four of them.

Answer: A

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