Question

AOB is the positive quadrant of the ellipse in which OA=a, OB=b. The area between the arc AB and the chord AB of the ellipse is

• A. $\frac { 1 } { 2 } a b ( \pi + 2 )$
• B.
• C. $\frac { 1 } { 4 } a b ( \pi - 2 )$
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

Required area

= $\frac { \mathrm { b } } { \mathrm { a } } \left[ \frac { 1 } { 2 } \mathrm { x } \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } } + \frac { 1 } { 2 } \mathrm { a } ^ { 2 } \sin ^ { - 1 } \frac { \mathrm { x } } { \mathrm { a } } \right] _ { 0 } ^ { \mathrm { a } } + \frac { \mathrm { b } } { 2 \mathrm { a } } \left[ ( \mathrm { a } - \mathrm { x } ) ^ { 2 } \right] _ { 0 } ^ { \mathrm { a } }$

= $\frac { b } { a } \left[ \frac { 1 } { 2 } a ^ { 2 } \sin ^ { - 1 } ( 1 ) \right] + \frac { b } { 2 a } \left( 0 - a ^ { 2 } \right)$