Area of the parallelogram whose sides are $x\cos \alpha + y\sin \alpha = p$ $x\cos \alpha + y\sin \alpha = q,\,\,$ $x\cos \beta + y\sin \beta = r$ and $x\cos \beta + y\sin \beta = s$ is

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $(0, 0), (0, 21)$ and $(21, 0)$, is

Triangle formed by the lines $3x + y + 4 = 0$ , $3x + 4y -15 = 0$ and $24x -7y = 3$ is a/an

A triangle is formed by $X -$ axis, $Y$ - axis and the line $3 x+4 y=60$. Then the number of points $P ( a, b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is $...........$

Equation of one of the sides of an isosceles right angled triangle whose hypotenuse is $3x + 4y = 4$ and the opposite vertex of the hypotenuse is $(2, 2)$, will be

The opposite angular points of a square are $(3,\;4)$ and $(1,\; - \;1)$. Then the co-ordinates of other two points are