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If the area enclosed between the curves ...

If the area enclosed between the curves `y=a x^2a n dx=a y^2(a >0)` is `1` square unit, then find the value of `adot`

A

`(1)/(sqrt(3))`

B

`(1)/(2)`

C

1

D

`(1)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
As from the figure, area enclosed between the curves is OABCO.
Thus, the point of intersection of
` y=ax^(2) " and " x=ay^(2)`

` rArr x=a(ax^(2))^(2) `
` rArr x=0,(1)/(a) rArr y=0,(1)/(a)`
So, the points of intersection are (0,0) and `((1)/(a),(1)/(a)).`
` therefore ` Required area OABCO= Area of curve OCBDO- Area of curve OABDO
` rArr int_(0)^(1//a)(sqrt((x)/(a))-ax^(2))dx=1 " "["given"]`
`rArr [(1)/(sqrt(a))*(x^(3//2))/(3//2)-(ax^(3))/(3)]_(0)^(1//a)=1 `
` rArr (2)/(3a^(2))-(1)/(3a^(2))=1`
` rArr a^(2)=(1)/(3) rArr a=(1)/(sqrt(3)) " "[ because a gt 0] `
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