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 and y)


Correct option is


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 < 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 = 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

           (x, y) = 2x2 + 3y2 – 12x + 12+ 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  ?


The in centre of the triangle with vertices , (0, 0) and (2, 0) is


If a vertex of a triangle is (1, 1) and the mid-point of two sides through the vertex are (–1, 2) and (3, 2), then the centroid of the triangle is


If the equation of the locus of a point equidistant from the points

(a1b1) and (a2b2) is (a1 – a1)x + (b1  –  b2)y + c = 0,

Then the value of c is :


The line joining the point  is produced to the point L(x, y) so that AL : LB b : a, then 



The vertex of a triangle are the points A(–36, 7), B(20, 7) and C(0, –8), If and I be the centoid and in center of the triangle, then GI is equal to


If a, b, c are relation by 4a2 + 9b2 – 9c2 + 12ab = 0 then the greatest distance between any two lines of the family of lines

ax + by + c = 0 is:


If x1x2x3 as well as y1y2y3 are in G.P. with the same common ratio, then the points (x1y1), (x2y2) and (x3y3)