Find the centre and radius of the circle 2x^2 + 2y^2- x = 0.

Question

Find the centre and radius of the circle $2x^2 + 2y^2- x = 0.$

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Answer

The given equation of circle is
$2x^2 + 2y^2 - x = 0 \Rightarrow x^2 + y^2 - \frac { x } { 2 } = 0$
$\Rightarrow \left( x ^ { 2 } - \frac { x } { 2 } \right)+ y^2 = 0$
On adding $\frac { 1 } { 16 }$ to make perfect squares, we get
$\left( x ^ { 2 } - \frac { x } { 2 } + \frac { 1 } { 16 } \right) + y^2 = \frac { 1 } { 16 }$
$\Rightarrow \left( x - \frac { 1 } { 4 } \right) ^ { 2 } + (y - 0)^2 = \left( \frac { 1 } { 4 } \right) ^ { 2 }$
On comparing with $(x - h)^2 + (y - k)^2 = r^2,$ we get
$h = \frac { 1 } { 4 }, k = 0$ and $r = \frac { 1 } { 4 }$
$\therefore$ Centre =$ (h, k) = \left( \frac { 1 } { 4 } , 0 \right)$
and Radius $= \frac { 1 } { 4 }$

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