Find the area of the circle `x^2+y^2=a^2` with radius `a`

Options

A. `pi a^(2)sq.` units
B. `pi a^(3)sq.` units
C. `pi a^(4)sq.` units
D. `pi a^(5)sq.` units

Answer

A

Explanation

`x^(2) +y^(2)=a^(2)`
The centre of this circle is (0, 0) and radius is 'a'. X- and Y- axes divide the circle into 4 symmetric parts.
Now, `x^(2)+y^(2)=a^(2)`
`implies y=+-sqrt(a^(2)-x^(2))`
In first quadrant,
` y=sqrt(a^(2)-x^(2))`

Co-ordinates of points O and A are (0, 0) and (a, 0) respectively. Now area of region bounded by the circle, X-axis, x = 0 and x = a
= Area of OABO
`=int_(0)^(a)"y "dx=int_(0)^(a)sqrt(a^(2)-x^(2))dx`
`=[(x)/(2)sqrt(a^(2)-x^(2))+(a^(2))/(2)"sin"^(-1)(x)/(a)]_(0)^(a)`
`=(0+(a^(2))/(2)"sin"^(-1)1)-(0+0)`
`=(a^(2)pi)/(2xx2)=(1)/(4)pia^(2)`
` :. ` Area of circle `= 4 xx `Area of OABO
`=4xx(1)/(4)pia^(2)=pi a^(2)sq.` units.