Question

The area of the region bounded by the curve a⁴y² = (2a – x) x⁵ is to that of the circle whose radius is a, is given by the ratio –

• A. 4 : 5
• B. 5 : 8
• C. 2 : 3
• D. 3 : 2

Answer 1

Answer: (B)

5 : 8

Answer 2

Given curve a⁴y² = (2a – x) x⁵

cutt off x-axis when y = 0 0 = (2a – x) x⁵ \ x = 0, 2aHence the area bounded by the curve

a⁴y² = (2a – x) x⁵ is

A₁ = $\int _ { 0 } ^ { 2 a } \frac { \sqrt { ( 2 a - x ) } } { a ^ { 2 } }$x^5/2 dx Put x = 2a sin² q \ dx = 4a sin q cos q dq \A₁ = dq = sin⁶q . cos²q dq

= 32a² . . $\frac { \pi } { 2 }$ = $\frac { 5 \pi \mathrm { a } ^ { 2 } } { 8 }$ (By walli's formula)

Area of circle A₂ = pa²

\ $\frac { \mathrm { A } _ { 1 } } { \mathrm {~A} _ { 2 } } = \frac { 5 } { 8 }$ Ž A₁ : A₂ = 5 : 8.