Triangle is formed by the coordinates (0, 0), (0, 21) and (21, 0). Find the numbers of integral coordinates strictly inside the triangle (integral coordinates of both x and y)
The three points from a right angled triangle whose hypotenuse by intercepts from is or x + y = 21. We have to find the number of points having integral coordinates and lying within the triangle. In other words we have to find the number of points having integral solution of the inequality x + y < 21 or x < 21 – y, where 0 < x, y < 21.
For x = 1, y = 20 but the point (1, 20) lies on the line AB and hence excluded. So for x = 1 we can have y = 1, 2,…...,19 (20 is excluded). Similarly, for x = 2, we can have y = 1, 2,… 18 and so on. Hence the total is
19 + 18 + 17 + … + 1
Determiner as to what point the axes of the coordinates be shifted so as to remove the first degree terms from the equation
f (x, y) = 2x2 + 3y2 – 12x + 12y + 24 = 0
What will be the coordinates of the point when the axes are rotated through an angle of 300 in clockwise sense?
What will be the coordinates of the point in original position ifr its coordinates after rotation of axes through an angle 600 ?
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