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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 ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}$.

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

• The equation of a circle with as the extremities of a diameter is .

• The equation of the circle with radius r, touching both the axes and lying in the first quadrant is ${\left(x-r\right)}^{2}+{\left(y-r\right)}^{2}={r}^{2}$.
•  The equation of the circle with centre (a, b) and touching the x-axis only is ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={b}^{2}$.
•  The equation of the circle with centre (a, b) and touching the y-axis only is ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={a}^{2}$.
• The parametric equation of the circle ${x}^{2}+{y}^{2}={r}^{2}$ is . Any point on the circle ${x}^{2}+{y}^{2}={r}^{2}$ is given by .
• The parametric equation of the circle ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}$ is . Any point on the circle ${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}$ is given by .

Let $S\equiv {x}^{2}+{y}^{2}+2gx+2fy+c=0$ be the general equation of a circle. Let …

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