Find the centre and radius of the circle 2×2 + 2y2 – x = 0
Question:

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

Solution:

The equation of the given circle is $2 x^{2}+2 y^{2}-x=0$.

$2 x^{2}+2 y^{2}-x=0$

$\Rightarrow\left(2 x^{2}-x\right)+2 y^{2}=0$

$\Rightarrow 2\left[\left(x^{2}-\frac{x}{2}\right)+y^{2}\right]=0$

$\Rightarrow\left\{x^{2}-2 x\left(\frac{1}{4}\right)+\left(\frac{1}{4}\right)^{2}\right\}+y^{2}-\left(\frac{1}{4}\right)^{2}=0$

$\Rightarrow\left(x-\frac{1}{4}\right)^{2}+(y-0)^{2}=\left(\frac{1}{4}\right)^{2}$ which is of the form $(x-h)^{2}+(y-k)^{2}=r^{2}$, where $h=\frac{1}{4}, k=0$, and $r=\frac{1}{4}$

Thus, the centre of the given circle is $\left(\frac{1}{4}, 0\right)$, while its radius is $\frac{1}{4}$.

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