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Let's do this question, find a number of points, having integral coordinates, not lying outside triangle

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ABC already, or pause the video, give it a try and then let's do it together.

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All right, let's do it together now.

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All right, so over here, it's mentioned that the Gordon points should have integral coordinates.

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That means X and Y values should be integers.

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OK, so it's given a zero 10 C's, 14 zero and B zero zero.

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OK, so if I extend these lines like this to form a rectangle right now, can you tell me in this rectangle

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how many integral coordinates will be there.

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You will have for 14 plus one.

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That's 15 points over here.

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Right.

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And you will have ten plus one.

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That's 11 points over here.

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Right.

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Zero one, two, three x a drop to ten.

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This is zero one, two, three x drop to 14.

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That's 15 points.

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We had 11 points.

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So in this rectangle, you will have a total of 11 and 15.

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That is one hundred and sixty five integral coordinates.

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That is positions where the X and Y value will be an integer value.

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All right.

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Now all here see is the diagonal, right?

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OK, so let's try to find the equation of this diagonal now, you know, two points, you know, zero,

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10 and 14 zero is a point, right, using the two point form, which is why minus Wyvern by X, minus

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X one is equal to Y two minus Y, one by X2, minus X one.

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Using that I can write the equation of X Y minus ten by X, minus zero is equal to zero, minus ten

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by 14, minus zero.

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All right.

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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

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to 140.

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All right.

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Now let's find the integral solutions for this equation.

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Do I have X over here and I have Y over here?

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When Y is equal to zero, I have ten, X is equal to 140.

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That is X is equal to 14.

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That's an integral solution.

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Now you will find that the next integral solution is possible when Y is equal to five, because at that

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time X is equal to seven.

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How do you do that now?

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If you take Y is equal to one, what do you get?

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10X minus the next is equal to one for T minus 14.

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Right.

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So over here you have zero.

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And if you subtract from here any value other than a multiple of ten.

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Right.

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You won't get another multiple often.

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For example, one for T minus 14.

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That's equal to one twenty six.

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Right.

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And that is not a multiple of ten.

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Right.

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Similarly if you do one for T minus two for it into two is twenty one for the minus 28 gives you a hundred

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and twelve again that's not a multiple of ten.

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So why is equal to one, two, three, four will not give you multiples of ten.

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So they are excluded at y equal to five.

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You get one or two to five.

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That's 70 right now.

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One for T minus seventy seventy then X is equal to seventy seven but again X seven and wi fi this is

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also an integral solution right now.

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What is the next integral solution possible.

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The next one is possible when Y is equal to ten.

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At that point you get X is equal to zero.

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So you can see that there are only three integral solutions along this diagonal.

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Right now, if I do one sixty five minus three, that is equal to one sixty two.

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Right.

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So that means if I take all the integral points on this side of the diagonal and all the integral points

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on this side of the diagonal in this rectangle excluding the diagonal, I have 162 points.

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That means one.

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Sixty two by two, that is eighty one points will be on this side and 81 points will be on the side.

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Now in this question, we need to find integral solutions not lying outside the triangle.

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Right.

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That means it's either inside or it's on the border.

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All right, so.

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So we have that there are 81 points on this site, which is not on the diagonal, so the total number

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of points which are inside are eighty one, right.

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Including the site.

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Right.

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It's including this site, the site and whatever is there available over here.

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Everything except what is on the diagonal.

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Now, the diagonal is also on the border.

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So we need to add that also to this 81.

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So our answer is eighty one plus these three point three, one point two point and three point eighty

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one plus three.

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That's equal to eighty four, which is the solution to this question.

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So that's another option A.