Question

If the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } }$ + $\frac { \mathrm { y } ^ { 2 } } { \mathrm {~b} ^ { 2 } }$ = 1 meets x-axis in A and y-axis in B in first quadrant, then area between the arc AB and the chord AB of the ellipse is –

• A. $\frac { 1 } { 2 }$ab (p + 2)
• B. $\frac { 1 } { 4 }$ ab (p – 2)
• C. $\frac { 1 } { 4 }$ ab (p – 4)
• D. None of these

Answer 1

Answer: (B)

$\frac { 1 } { 4 }$ ab (p – 2)

Answer 2

$\frac { x ^ { 2 } } { a ^ { 2 } }$ + = 1 … (1) Ž y = ±

Equation of chord AB is y – 0 = $\frac { b - 0 } { 0 - a }$ (x – a) or

y = – (x – a) … (2)

\ Required

Area = $\int _ { 0 } ^ { a } \left\{ \frac { b } { a } \sqrt { a ^ { 2 } - x ^ { 2 } } - \left( - \frac { b } { a } \right) ( x - a ) \right\} d x$

=

= (p – 2). $\left( \because \sin ^ { - 1 } 1 = \frac { \pi } { 2 } \right)$