Co Ordinate Geometry

Step 1: Consider a cone with a base radius and height . A cylinder inscribed in the cone has a radius and height .

Step 2: The volume of the cylinder is maximized when its surface area is maximized. The curved surface area of the cylinder is given by: where and depend on the geometry of the cone.

Step 3: Using the geometric relations between the cone's dimensions and the cylinder's dimensions, you can show that the radius of the cylinder at maximum surface area is . Thus, the radius of the cylinder of maximum curved surface is half the radius of the cone.

Step 1: The condition for the perpendicularity of two lines is that the dot product of their direction ratios is zero. The direction ratios of the first line are , and the direction ratios of the second line are .

Step 2: The dot product of these direction ratios is: Thus, the value of is .