Conic Sections - AP Precalculus

$y = \pm \frac{b}{a}x$. The slopes are $\pm\frac{b}{a}$ for the standard hyperbola form.

$c = \sqrt{a^2 - b^2}$. For an ellipse, $c < a$ since the foci are inside the ellipse.

Circle. Complete the square to verify equal coefficients for $x^2$ and $y^2$.

$x^2 + y^2 = r^2$. Where $r$ is the radius from the center at $(0,0)$.

$|4p|$. The chord through the focus perpendicular to the axis of symmetry.

Hyperbola. Difference of squares with equal coefficients indicates a hyperbola.

(4, 0). From $y^2 = 16x$, we get $4p = 16$, so $p = 4$ and focus is at $(p,0)$.

Center: (0,0), $a=2$, $b=3$. Divide by 36: $\frac{x^2}{4} + \frac{y^2}{9} = 1$, so $a^2 = 4$, $b^2 = 9$.

$|4p|$. The chord through the focus perpendicular to the axis of symmetry.

Circle. Complete the square to verify it has equal coefficients for squared terms.

$c^2 = a^2 + b^2$. For hyperbolas, $c > a$ since the foci are outside the vertices.

$y = -2$. From $x^2 = 8y$, we have $4p = 8$, so $p = 2$ and directrix is $y = -p$.

$(x-h)^2 + (y-k)^2 = r^2$. Standard form for any circle with center and radius specified.

Circle. Complete the square to verify it forms a circle equation.

Hyperbola. Eccentricity greater than 1 distinguishes hyperbolas from other conics.

A set of points equidistant from a point (focus) and a line (directrix). This property defines the parabola's unique geometric shape.

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. The discriminant $B^2 - 4AC$ determines the conic type.

$x^2 = 4py$. Where $p$ is the distance from vertex to focus and directrix.

$e = \frac{c}{a}$ where $c = \sqrt{a^2 - b^2}$. For ellipses, $0 < e < 1$ since $c < a$ always.

Parabola. Square term on $y$ with linear $x$ indicates a horizontal parabola.

(2, 5). The vertex form shows the vertex at $(h,k)$ where the parabola turns.

$x^2 + y^2 = r^2$. Where $r$ is the radius from the center at $(0,0)$.

A set of points where the sum of distances to two foci is constant. The sum equals $2a$, where $a$ is the semi-major axis length.

$y = ax^2$. Where $a$ determines the width and opens vertically.

Circle. Complete the square to verify it has equal coefficients for $x^2$ and $y^2$ terms.

$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Derived from the Pythagorean theorem in coordinate geometry.

A set of points where the difference of distances to two foci is constant. The absolute value of the difference equals $2a$, where $a$ is the semi-major axis.

Hyperbola. Difference of squares form indicates a hyperbola with $a^2 = b^2 = 1$.

$e = \frac{c}{a}$. Where $c$ is the focal distance and $a$ is the semi-major axis.