The equation (x^2 + y^2 = r^2) represents a circle with radius (r) and center at the origin.

The eccentricity of a circle is 0 because it is a perfectly round shape with no elongation.

The equation (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1) represents an ellipse with semi-major axis (a) and semi-minor axis (b).

The eccentricity of an ellipse is a value between 0 and 1 that determines how elongated the ellipse is.

The equation (y^2 = 4px) represents a parabola with vertex at the origin and axis of symmetry along the (x)-axis.

The eccentricity of a parabola is undefined because it is an open curve that extends infinitely.

The equation (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1) represents a hyperbola with transverse axis along the (x)-axis and conjugate axis along the (y)-axis.

The eccentricity of a hyperbola is a value greater than 1 that determines how elongated the hyperbola is.

The standard form of the equation of a circle is (x^2 + y^2 = r^2), where (r) is the radius of the circle.

The standard form of the equation of an ellipse is (\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the semi-major and semi-minor axes, respectively.

The standard form of the equation of a parabola is (y^2 = 4px), where (p) is the distance from the vertex to the focus.

The standard form of the equation of a hyperbola is (\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1), where (a) and (b) are the lengths of the transverse and conjugate axes, respectively.

The equation of the directrix of a parabola with vertex at the origin and focus at ((p, 0)) is (x = -p).

The equation of the directrix of an ellipse with center at the origin and semi-major axis (a) is (x = a).

The equation of the directrix of a hyperbola with center at the origin and transverse axis (2a) is (x = 2a).