Circle & Conics
 A circle is the locus of a point which moves in such a way that its distance from a fixed point is constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

The equation of the circle with radius r and centre (0, 0) is ${x}^{2}+{y}^{2}={r}^{2}$.

The equation of the circle with centre (a, b) and radius r is ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={r}^{2}$.

General equation of the circle is ${x}^{2}+{y}^{2}+2gx+2fy+c=0$, where $\left(g,f\right)$ is the centre and $r=\sqrt{{g}^{2}+{f}^{2}c}$ is the radius of the circle.

The equation of a circle with $A\left({x}_{1},{y}_{1}\right)\mathrm{and}B\left({x}_{2},{y}_{2}\right)$ as the extremities of a diameter is $\left(x{x}_{1}\right)\left(x{x}_{2}\right)+\left(y{y}_{1}\right)\left(y{y}_{2}\right)=0$.
 The equation of the circle with radius r, touching both the axes and lying in the first quadrant is ${\left(xr\right)}^{2}+{\left(yr\right)}^{2}={r}^{2}$.
 The equation of the circle with centre (a, b) and touching the xaxis only is ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={b}^{2}$.
 The equation of the circle with centre (a, b) and touching the yaxis only is ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={a}^{2}$.
 The parametric equation of the circle ${x}^{2}+{y}^{2}={r}^{2}$ is $x=r\mathrm{cos}\theta ,y=r\mathrm{sin}\theta ;0\le \theta \le 2\pi $. Any point on the circle ${x}^{2}+{y}^{2}={r}^{2}$ is given by $\left(r\mathrm{cos}\theta ,r\mathrm{sin}\theta \right)$.
 The parametric equation of the circle ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={r}^{2}$ is $x=a+r\mathrm{cos}\theta ,y=b+r\mathrm{sin}\theta ;0\le \theta \le 2\pi $. Any point on the circle ${\left(xa\right)}^{2}+{\left(yb\right)}^{2}={r}^{2}$ is given by $\left(a+r\mathrm{cos}\theta ,b+r\mathrm{sin}\theta \right)$.
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