Let's do this question, find a number of points, having integral coordinates, not lying outside triangle
ABC already, or pause the video, give it a try and then let's do it together.
All right, let's do it together now.
All right, so over here, it's mentioned that the Gordon points should have integral coordinates.
That means X and Y values should be integers.
OK, so it's given a zero 10 C's, 14 zero and B zero zero.
OK, so if I extend these lines like this to form a rectangle right now, can you tell me in this rectangle
how many integral coordinates will be there.
You will have for 14 plus one.
That's 15 points over here.
Right.
And you will have ten plus one.
That's 11 points over here.
Right.
Zero one, two, three x a drop to ten.
This is zero one, two, three x drop to 14.
That's 15 points.
We had 11 points.
So in this rectangle, you will have a total of 11 and 15.
That is one hundred and sixty five integral coordinates.
That is positions where the X and Y value will be an integer value.
All right.
Now all here see is the diagonal, right?
OK, so let's try to find the equation of this diagonal now, you know, two points, you know, zero,
10 and 14 zero is a point, right, using the two point form, which is why minus Wyvern by X, minus
X one is equal to Y two minus Y, one by X2, minus X one.
Using that I can write the equation of X Y minus ten by X, minus zero is equal to zero, minus ten
by 14, minus zero.
All right.
This simplifies to 14 y minus 140 is equal to minus 10 X or I can write it as 14 Y plus then X is equal
to 140.
All right.
Now let's find the integral solutions for this equation.
Do I have X over here and I have Y over here?
When Y is equal to zero, I have ten, X is equal to 140.
That is X is equal to 14.
That's an integral solution.
Now you will find that the next integral solution is possible when Y is equal to five, because at that
time X is equal to seven.
How do you do that now?
If you take Y is equal to one, what do you get?
10X minus the next is equal to one for T minus 14.
Right.
So over here you have zero.
And if you subtract from here any value other than a multiple of ten.
Right.
You won't get another multiple often.
For example, one for T minus 14.
That's equal to one twenty six.
Right.
And that is not a multiple of ten.
Right.
Similarly if you do one for T minus two for it into two is twenty one for the minus 28 gives you a hundred
and twelve again that's not a multiple of ten.
So why is equal to one, two, three, four will not give you multiples of ten.
So they are excluded at y equal to five.
You get one or two to five.
That's 70 right now.
One for T minus seventy seventy then X is equal to seventy seven but again X seven and wi fi this is
also an integral solution right now.
What is the next integral solution possible.
The next one is possible when Y is equal to ten.
At that point you get X is equal to zero.
So you can see that there are only three integral solutions along this diagonal.
Right now, if I do one sixty five minus three, that is equal to one sixty two.
Right.
So that means if I take all the integral points on this side of the diagonal and all the integral points
on this side of the diagonal in this rectangle excluding the diagonal, I have 162 points.
That means one.
Sixty two by two, that is eighty one points will be on this side and 81 points will be on the side.
Now in this question, we need to find integral solutions not lying outside the triangle.
Right.
That means it's either inside or it's on the border.
All right, so.
So we have that there are 81 points on this site, which is not on the diagonal, so the total number
of points which are inside are eighty one, right.
Including the site.
Right.
It's including this site, the site and whatever is there available over here.
Everything except what is on the diagonal.
Now, the diagonal is also on the border.
So we need to add that also to this 81.
So our answer is eighty one plus these three point three, one point two point and three point eighty
one plus three.
That's equal to eighty four, which is the solution to this question.
So that's another option A.