Conic Sections - Circle

A circle is defined as the set of all points in a plane that are at a fixed distance, known as the radius $r$, from a fixed point, known as the center $(h, k)$. Visually, the circle forms a perfectly round closed loop on the Cartesian plane where every point on the perimeter is equidistant from the midpoint.

The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$. This formula is derived from the distance formula; geometrically, it represents the locus of a point $P(x, y)$ such that its distance from the center $C(h, k)$ is always $r$.

When the center of the circle is at the origin $(0, 0)$, the equation simplifies to $x^2 + y^2 = r^2$. Visually, the circle is centered at the intersection of the $x$ and $y$ axes, showing symmetry across both axes.

The general equation of a circle is expressed as $x^2 + y^2 + 2gx + 2fy + c = 0$. For an equation to represent a circle, the coefficients of $x^2$ and $y^2$ must be equal and there should be no $xy$ term. Geometrically, the center of this circle is located at $(-g, -f)$.

To find the radius $r$ from the general form, we use the expression $\sqrt{g^2 + f^2 - c}$. Visually, for a circle to be 'real', the value inside the square root ($g^2 + f^2 - c$) must be positive. If it is zero, the circle collapses into a single point (point circle).

A circle touching the axes has special geometric properties. If a circle touches the $x$-axis, its radius is equal to the absolute value of the $y$-coordinate of its center ($r = |k|$). If it touches the $y$-axis, the radius is equal to the absolute value of the $x$-coordinate ($r = |h|$).

The position of a point $(x_1, y_1)$ relative to a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is determined by the sign of $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$. Visually, if $S_1 < 0$, the point is inside the circle; if $S_1 = 0$, it is on the boundary; and if $S_1 > 0$, it lies outside.