Question

The maximum area of the rectangle that can be inscribed in the circle given by x = 3 + 5 cos θ, y = 1 + 5 sin θ in sq. Units is

Answer 1

Given:

x = 3+5cosθ

y = 1+5sinθ

To find:

The maximum area of the rectangle that can be inscribed in the circle

Calculation:

The radius of the given circle is(r) = 5

Let a,b be the length and breadth of the rectangle.The length diagonals of the rectangle will be equal to the diameter of the circle

   $\sqrt{a^2+b^2} =10$

      $a^2+b^2=100$

The area of the rectangle = a×b

From the relation

      $(a-b)^2\geq 0$

      $(a^2+b^2)-2ab\geq 0$

      $100-2ab\geq 0$

      $2ab\leq100$

      $ab\leq 50$

Final answer:

The maximum area of the rectangle that can be inscribed in the circle is 50

sq.units