UT m408D conic section formulas

Circle standard form

$$(x − h)^2 + (y − k)^2 = r^2$$

Circle center

$$(h, k)$$

Circle radius

$$r$$

Horizontal ellipse

$$\frac{(x − h)^2}{a^2} + \frac{(y − k)^2}{b^2} = 1$$

Vertical ellipse

$$\frac{(x − h)^2}{b^2} + \frac{(y − k)^2}{a^2} = 1$$

Ellipse foci relation

$$c^2 = a^2 − b^2$$

Ellipse eccentricity

$$e = \frac{c}{a}$$

Horizontal ellipse foci

$$(h ± c, k)$$

Vertical ellipse foci

$$(h, k ± c)$$

Ellipse parametric equations

$$x = a \cos(t), \ y = b \sin(t)$$

Ellipse distance property

$$L_1 + L_2 = 2a$$

Horizontal hyperbola

$$\frac{(x − h)^2}{a^2} − \frac{(y − k)^2}{b^2} = 1$$

Vertical hyperbola

$$\frac{(y − k)^2}{a^2} − \frac{(x − h)^2}{b^2} = 1$$

Hyperbola foci relation

$$c^2 = a^2 + b^2$$

Hyperbola eccentricity

$$e = \frac{c}{a}$$

Horizontal hyperbola foci

$$(h ± c, k)$$

Vertical hyperbola foci

$$(h, k ± c)$$

Horizontal hyperbola asymptotes

$$y − k = ±\frac{b}{a}(x − h)$$

Vertical hyperbola asymptotes

$$y − k = ±\frac{a}{b}(x − h)$$

Vertical parabola

$$(x − h)^2 = 4p(y − k)$$

Horizontal parabola

$$(y − k)^2 = 4p(x − h)$$

Vertical parabola focus

$$(h, k + p)$$

Vertical parabola directrix

$$y = k − p$$

Horizontal parabola focus

$$(h + p, k)$$

Horizontal parabola directrix

$$x = h − p$$

General conic equation

$$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0$$

Completing the square

$$x^2 + bx = \left(x + \frac{b}{2}\right)^2 − \left(\frac{b}{2}\right)^2$$

Circle identification

Same squared coefficients and same signs.

Ellipse identification

Same signs, different coefficients.

Hyperbola identification

Opposite signs on squared terms.

Parabola identification

Only one squared variable.