Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3r`.

The equation of a circle of radius r and touching both the axes is

Find the dimensions of the largest rectangle that can be inscribed in a semi circle of radius r cm.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is (4r)/(3) .

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

A stone is fastened to one end of a string and is whirled in a verticla circle of radius R. Find the minimum speed the stone can have at the highest point of the circle.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is (2R)/(sqrt(3)) . Also find the maximum volume.

Equation of the straight line that forms are isosceles triangle with coordinate axes in the I quadrant with perimeter 4+2sqrt(2) is ______