Plot the region in the first quadrant in which points are nearer to the origin than to the line `x=3.`

Consider a square with vertices at (1,1)(-1,1)(-1,-1)and(1,-1). Let S be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region S and find its area.

Find the point on the line 2x+3y =6 which is closest to the origin.

Find the point on the straight line 3x-4y=25,nearest to the origin .

Find the equation of the straight line which passes through the origin and makes angle 60^0 with the line x+sqrt(3)y+sqrt(3)=0 .

Find the area of the region in the first quadrant enclosed by the x-aixs, the line y = x, and the circle x^(2) + y^(2) = 32 .

Find the area of the region in the first quadrant enclosed by x-axis, line x = sqrt3y and the circle r^(2)+ y^(2) = 4 .

The number of integral points (x,y) (i.e, x and y both are integers) which lie in the first quadrant but not on the coordinate axes and also on the straight line 3x + 5y = 2007 is equal to (A) 133 (B) 135 (C) 138 (D) 140

The curve in the first quadrant for which the normal at any point (x , y) and the line joining the origin to that point form an isosceles triangle with the x-axis as base is (a) an ellipse (b) a rectangular hyperbola (c) a circle (d) None of these

Find the area lying in the first quadrant and bounded by the curve y=x^3 and the line y=4xdot

OABCDE is a regular hexagon of side 2 units in the XY-plane in the first quadrant. O being the origin and OA taken along the x-axis. A point P is taken on a line parallel to the z-axis through the centre of the hexagon at a distance of 3 unit from O in the positive Z direction. Then find vector AP.