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Question

The area enclosed by the curvesy+x1, y-x1 and 2x2+2y2=1 is


A

2+π2squnits

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B

2-π2squnits

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C

3+π2squnits

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D

3-π4squnits

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Solution

The correct option is B

2-π2squnits


Explanation for correct answer Option(s)

Finding the area enclosed by the curves.

The given Equation of the curves are,

y+x1......(1)

y-x1......(2)

The Equation of a circle is

2x2+2y2=1

Rewrite the above Equation as,

2x2+y2=1x2+y2=12x2+y2=122

Here the radius of a circle is r=12 with center 0,0

The graph is as shown:

The Equation (1) and Equation (2) from a square ABCD

y+x=1

Put x=0 we get

y+0=1y=1

Put y=0 we get

0+x=1x=1

Hence, the points are 0,11,0

Then,

y-x=1

Put x=0 we get

y-0=1y=1

Put y=0 we get

0-x=1x=-1

Hence, the points are 0,1-1,0

From the graph BC is equal to

BC=0-12+1-02BC=1+1BC=2

Also From the graph DC is equal to

DC=0+12+1-02DC=1+1DC=2

Now area of a square ABCD is Equal to

DC×BC=2×2=2squnits

Now the area of a shaded region is Equal to

A=Areaofsquare-AreaofcircleA=2-πr2A=2-π122A=2-π2squnits

Therefore, the correct answer is Option B.


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