Conic Sections - Circles, Semicircles, Ellipses, Hyperbolas, and Parabolas

The equation of the directrix for a parabola that opens up or down is y = k - p.

The standard form of the equation of a circle is (x - h)² + (y - k)² = r².

The eccentricity of a circle is zero.

The area of a circle is πr².

The circumference of a circle is 2πr.

The domain of a circle is [h - r, h + r].

The range of a circle is [k - r, k + r].

By solving for y in the circle equation, you can get y = ±√(r² - (x - h)²) + k.

The equation for the ellipse with a horizontal major axis is (x - h)²/a² + (y - k)²/b² = 1.

The equation for the ellipse with a vertical major axis is (x - h)²/b² + (y - k)²/a² = 1.

The eccentricity of an ellipse is c/a, where c is the distance from the center to a focus and a is the semi-major axis.

The area of an ellipse is πab, where a and b are the semi-major and semi-minor axes, respectively.

The coordinates of the foci of an ellipse are (h ± c, k), where c is the distance from the center to a focus.

The equation for the hyperbola with a horizontal transverse axis is (x - h)²/a² - (y - k)²/b² = 1.

The equation for the hyperbola with a vertical transverse axis is (y - k)²/a² - (x - h)²/b² = 1.

The eccentricity of a hyperbola is c/a, where c is the distance from the center to a focus and a is the semi-major axis.

The coordinates of the foci of a hyperbola are (h ± c, k) for the horizontal transverse axis and (h, k ± c) for the vertical transverse axis.

The equation of the asymptotes for a hyperbola with a horizontal transverse axis is y = ±(b/a)(x - h) + k.

The equation of the asymptotes for a hyperbola with a vertical transverse axis is y = ±(a/b)(x - h) + k.

The x-intercepts of an ellipse with a horizontal major axis are (±a, 0).

The x-intercepts of an ellipse with a vertical major axis are (0, ±b).

The x-intercepts of a hyperbola with a horizontal transverse axis are (±a, 0).

The x-intercepts of a hyperbola with a vertical transverse axis are (0, ±b).

The eccentricity of a parabola is one.

The coordinates of the focus of a parabola that opens to the right or left are (h + p, k).

The coordinates of the focus of a parabola that opens up or down are (h, k + p).

The equation of the directrix for a parabola that opens to the right or left is x = h - p.

The equation of the directrix for a parabola that opens up or down is y = k - p.

The standard form of the equation of a circle is (x - h)² + (y - k)² = r².