Rectangles are inscribed inside a semi-circle of radius `rdot` Find the rectangle with maximum area.

Regular pentagons are inscribed in two circles of radius 5 and 2 units respectively. The ratio of their areas is

A triangle is inscribed in a circle of radius 1. The distance between the orthocentre and the circumcentre of the triangle cannot be

Prove that an angle inscribed in a semi-circle is a right angle using vector method.

The largest area of the trapezium inscribed in a semi-circle or radius R , if the lower base is on the diameter, is (a) (3sqrt(3))/4R^2 (b) (sqrt(3))/2R^2 (c) (3sqrt(3))/8R^2 (d) R^2

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

A rectangle is inscribed in an equilateral triangle of side length 2a units. The maximum area of this rectangle can be sqrt(3)a^2 (b) (sqrt(3)a^2)/4 a^2 (d) (sqrt(3)a^2)/2

ABC is an isosceles triangle inscribed in a circle of radius rdot If A B=A C and h is the altitude from A to B C , then triangle A B C has perimeter P=2(sqrt(2h r-h^2)+sqrt(2h r)) and area A= ____________ and also ("lim")_(h->0)A/(P^3)=______

Consider the ellipse whose major and minor axes are x-axis and y-axis, respectively. If phi is the angle between the CP and the normal at point P on the ellipse, and the greatest value tan phi is 3/4 (where C is the centre of the ellipse). Also semi-major axis is 10 units . A rectangle is inscribed in a ellipse whose sides are parallel to the coordinate axes, then the maximum area of rectangle is

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.