RD Sharma - Mathematics (Volume 2)

Book: RD Sharma - Mathematics (Volume 2)

Chapter: 21. Areas of Bounded Regions

Subject: Maths - Class 12th

Q. No. 17 of Exercise 21.3

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17

Prove that the area in the first quadrant enclosed by the axis, the line x = √3y and the circle x2 + y2 = 4 is π/3.

To find an area in the first quadrant enclosed by the x – axis,

x = √3y


x2 + y2 = 4


Or


Or


Or


And


Equation (i) represents a line passing through (0,0), ( – √3, – 1), (√3,1).


Equation (ii) represents a circle centre (0,0) and passing through (±2,0), (0,±2).


Points of intersection of line and circle are ( – √3, – 1) and (√3,1).


These are shown in the graph below: -



Required enclosed area = Region OABO


= Region OCBO + Region ABCA








Hence proved that the area in the first quadrant enclosed by the axis, the line and the circle is π/3.


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