Shapes formed when a plane intersects a cone.

All points equidistant from a focus and a directrix.

The turning point of the parabola.

A conic section where the sum of the distances from any point on the ellipse to two foci is constant.

Two special points inside the ellipse used in its definition.

A conic section formed by the intersection of a plane with a double cone.

Lines that the hyperbola approaches as it extends to infinity.

A line such that every point on the parabola is equidistant from the focus and the directrix.

The longest diameter of the ellipse, passing through the center and both foci.

The shortest diameter of the ellipse, passing through the center and perpendicular to the major axis.

Check for squared terms, their coefficients, and the sign between them. No squared term = line. Same coefficients with + = circle. Different coefficients with + = ellipse. - sign = hyperbola. One squared term = parabola.

Identify 'h' and 'k' from the standard equation $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$. The center is (h, k).

1. Find a and b. 2. Calculate c using $c^2 = a^2 + b^2$. 3. The foci are at (h ± c, k) if horizontal, or (h, k ± c) if vertical.

Identify 'h' and 'k' from the standard equation $(y - k)^2 = a(x - h)$ or $(x - h)^2 = a(y - k)$. The vertex is (h, k).

If the x-term is squared, it opens up or down. If the y-term is squared, it opens left or right. If the coefficient 'a' is positive, it opens up or right; if negative, down or left.

1. Find the center (midpoint of foci). 2. Find 'a' (half major axis length). 3. Find 'c' (distance from center to focus). 4. Calculate 'b' using $b^2 = a^2 - c^2$. 5. Write the equation.

For a hyperbola centered at (h, k) with equation $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, the asymptotes are $y - k = \pm \frac{b}{a}(x - h)$.

Complete the square for both x and y terms to get the equation into the standard form for the specific conic section.

1. Find the vertex (h, k). 2. Determine 'a' from the equation. 3. The directrix is a line y = k - p for parabolas opening up/down or x = h - p for parabolas opening left/right, where p = 1/(4a).

Identify 'a' and 'b' in the equation $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$. The major axis has length 2a, and the minor axis has length 2b.

Ellipse: '+' sign between squared terms, closed curve. | Hyperbola: '-' sign, two separate curves.

Circle: $(x-h)^2 + (y-k)^2 = r^2$, equal radii. | Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, different horizontal and vertical radii.

Parabola: One focus and directrix, one squared term. | Hyperbola: Two foci, two squared terms with a minus sign.

Ellipse: Foci are inside the curve, sum of distances to foci is constant. | Hyperbola: Foci are outside the curve, difference of distances to foci is constant.

Ellipse: 'a' is the semi-major axis length. | Hyperbola: 'a' is the distance from the center to the vertices along the transverse axis.

Ellipse: $c^2 = a^2 - b^2$. | Hyperbola: $c^2 = a^2 + b^2$.

Circle: The center is equidistant from all points on the circle. | Hyperbola: The center is the midpoint between the vertices and foci, but not on the hyperbola itself.

Hyperbola: Has asymptotes that guide the branches of the curve. | Ellipse: Does not have asymptotes.

Parabola: Determined by which variable is squared and the sign of the leading coefficient. | Hyperbola: Determined by which term (x or y) is positive in the standard equation.

Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. | Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.