An ellipse inscribed in a semi-circle touches the circular are at two distinct points and also touches the bounding diameter. its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area, its eccentricity is

Step-by-step solution

Let the equation of the ellipse be x² a² + y² b² =1 x²=a²- a² y² b² . (i) The equation of circle will be x²+(y-b)²=1 ....... (ii) Using (i) in (ii), we get array l a²- a² y² b² +(y-b)²=1 (1- a² b² ) y²+2 b y+a²+b²-r²=0 array As you can see from the figure the ellipse touches the circle for same value of y so the roots of the eqaution are equal D =04 ~b ²-4 (1- a ² ~b ² ) ( a ²+ b ²- r ² )=0 r ²= a ⁴ a ²- b ² b = a 1- a ² r ² Area of ellipse = ab A = a a 1- a ² r ² As area of ellipse is maximun array l d A d a =0 a