A line passes through A(1,1) and B(100,1000). How many other points wi

Step 1: find the equation of the line satisfying the 2 points

Slope = m = (1,000 - 1) / (100 - 1) = 999/99 = 111/11

Y = (111/11) X + B

Plugging in the coordinates (1 , 1) which fall on the line we find that the

The Y intercept = B = (-)(100/11)

Equation of given line is:

Y = (111/11)X - (100/11)

—multiply both sides by 11 and you have the Linear Indeterminate Equation of:

11Y + 100 = 111X

We are asked for the Integer Coordinate Pairs of (X , Y) that fall in between the 2 given points (1 , 1) and (100 , 1,000)


Step 2: we know that when X = 1 and Y = 1 the equation of: 11y + 100 = 111x : will be satisfied

Concept: because the variables appear on each side of the equation, the Corresponding Integer Pairs that Satisfy the Equation will follow this pattern

When you have a linear indeterminate equation such as:

(1)
a(X) + b(Y) = c

Where a and b and c = constant number values and X and Y = unknown Integer Values

(1st) find the first integer X, Y pair that satisfies the equation. Then each consecutive integer pair that satisfies the equation can be found by:

X + b (add the coefficient in front of the Y unknown variable)

And

Y - (a) (subtract the coefficient in front of the X unknown variable)

(2)
for a linear indeterminate equation of the form:

a(X) + c = b(Y)

find each consecutive pair of Integer (X , Y) values that satisfy the equation by:

X + b (add the coefficient in front of the Y variable)

And

Y + a (add the coefficient in front of the X variable)


For:

11y + 100 = 111x

First integer pair is (1 , 1)

X = 1 ———> each consecutive X Integer Value that satisfies the equation will increase by the Coefficient in front of the Y variable —-> (+11)

Y = 1 ———-> each consecutive corresponding Y Integer value that satisfies the equation will increase by the Coefficient in front of the X variable (+111)


Thus, the next pair of Integer coordinate values that will satisfy the equation:

Start: (X = 1) and (Y = 1)

(X= 12) and (Y = 112)

Then: add (+11) to the X value again —-and —- and add (+111) to the Y value again

(X = 23) and (Y = 123)

....

This will continue until

(X = 100) and (Y = 1,000)

We can use the X integer values to determine how many Pairs of (X, Y) integer values satisfy the equation:

We need to count the integer values in between [1 thru 100], NON- inclusive, when the Increment or spacing between each consecutive value is (+11)

Non-inclusive Count = [ (100 - 1) / 11 ] - 1

= (99/11) - 1

= 9 - 1


8

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All points between A and B will have the same slope as that of the given points.

Slope: \(\frac{1000 - 1 }{ 100 - 1} = \frac{999 }{ 99} = \frac{111}{ 11}\). This is \(\frac{100 + 11 }{ 11}\).

All points that will have the same slope must be expressed in terms of \(111/11\)

=> \(\frac{888}{88} , \frac{777}{77} , \frac{666}{66} , \frac{555}{55}, \frac{ 444}{44}, \frac{333}{33}, \frac{222}{22}, \frac{111}{11}\)


Eight points

Answer D

The points on the line are from (1,1) to (100,1000=111/11
Slope must be same for all 111/11
Since x and y are integers and they can all take the values
888/88,777/77,666/66,555/55,444/44,333/33,222/22,111/11
total 8 values.
Hence IMO D

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