Condition for real circle: $g^2+f^2-c > 0$

Diameter Form: $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$ (endpoints $(x_1, y_1), (x_2, y_2)$)

Equation of Tangent to $x^2+y^2=r^2$ at $(x_1, y_1)$: $xx_1 + yy_1 = r^2$
Equation of Tangent to $x^2+y^2+2gx+2fy+c=0$ at $(x_1, y_1)$: $xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$
Tangent in Slope Form to $x^2+y^2=r^2$: $y = mx \pm r\sqrt{1+m^2}$
Length of Tangent from $(x_1, y_1)$ to $x^2+y^2+2gx+2fy+c=0$: $\sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c}$ (denoted as $\sqrt{S_1}$)
Equation of Chord of Contact: $T=0$ (same as tangent equation, but for external point)
Equation of Normal: Line passing through center and point of tangency.

Equation of Chord with Midpoint $(x_1, y_1)$: $T=S_1$
Power of a Point $(x_1, y_1)$ w.r.t Circle $S=0$: $S_1 = x_1^2+y_1^2+2gx_1+2fy_1+c$
Radical Axis of two circles $S_1=0, S_2=0$: $S_1 - S_2 = 0$. Perpendicular to line joining centers.
Radical Centre: Intersection of radical axes of three circles.

Definition: Locus of a point equidistant from a fixed point (focus) and a fixed line (directrix).
Standard Equation: $y^2 = 4ax$
- Focus: $(a, 0)$
- Directrix: $x = -a$
- Vertex: $(0, 0)$
- Axis: $y = 0$ (x-axis)
- Latus Rectum Length: $4a$
Other Forms:
- $y^2 = -4ax$: Focus $(-a, 0)$, Directrix $x=a$
- $x^2 = 4ay$: Focus $(0, a)$, Directrix $y=-a$
- $x^2 = -4ay$: Focus $(0, -a)$, Directrix $y=a$
Parametric Equations for $y^2=4ax$: $(at^2, 2at)$