Conic Sections Formulas — Circle, Ellipse, Parabola, Hyperbola A complete reference of conic sections formulas: circles, ellipses, parabolas, and hyperbolas. Includes standard equation forms, foci, vertices, directrices, asymptotes, and eccentricity for each conic.
Circle x 2 + y 2 = r 2 x^2 + y^2 = r^2 x 2 + y 2 = r 2 ( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2 General Form
x 2 + y 2 + D x + E y + F = 0 x^2 + y^2 + D x + E y + F = 0 x 2 + y 2 + D x + E y + F = 0 Center from General Form
( h , k ) = ( − D 2 , − E 2 ) (h, k) = \left(-\tfrac{D}{2},\ -\tfrac{E}{2}\right) ( h , k ) = ( − 2 D , − 2 E ) Radius from General Form
r = h 2 + k 2 − F r = \sqrt{h^2 + k^2 - F} r = h 2 + k 2 − F Ellipse (Horizontal Major Axis) Standard Form
( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 ( a > b ) \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad(a > b) a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 ( a > b ) Vertices
( h ± a , k ) (h \pm a,\ k) ( h ± a , k ) Co-Vertices
( h , k ± b ) (h,\ k \pm b) ( h , k ± b ) Foci Distance
c = a 2 − b 2 c = \sqrt{a^2 - b^2} c = a 2 − b 2 Foci
( h ± c , k ) (h \pm c,\ k) ( h ± c , k ) Eccentricity
e = c a , 0 < e < 1 e = \frac{c}{a},\ 0 < e < 1 e = a c , 0 < e < 1 Ellipse (Vertical Major Axis) Standard Form
( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 ( a > b ) \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad(a > b) b 2 ( x − h ) 2 + a 2 ( y − k ) 2 = 1 ( a > b ) Vertices
( h , k ± a ) (h,\ k \pm a) ( h , k ± a ) Foci
( h , k ± c ) , c = a 2 − b 2 (h,\ k \pm c),\ c = \sqrt{a^2 - b^2} ( h , k ± c ) , c = a 2 − b 2 Parabola (Vertex Form) Opens Up/Down
y − k = a ( x − h ) 2 y - k = a(x - h)^2 y − k = a ( x − h ) 2 Opens Right/Left
x − h = a ( y − k ) 2 x - h = a(y - k)^2 x − h = a ( y − k ) 2 Focus (opens up, p = 1/(4a))
( h , k + p ) (h,\ k + p) ( h , k + p ) Standard Focus-Directrix (vertex at origin, opens up)
Axis of Symmetry
x = h (opens up/down) x = h \text{ (opens up/down)} x = h (opens up/down) Hyperbola (Horizontal Transverse Axis) Standard Form
( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 Vertices
( h ± a , k ) (h \pm a,\ k) ( h ± a , k ) Foci Distance
c = a 2 + b 2 c = \sqrt{a^2 + b^2} c = a 2 + b 2 Foci
( h ± c , k ) (h \pm c,\ k) ( h ± c , k ) Asymptotes
y − k = ± b a ( x − h ) y - k = \pm\tfrac{b}{a}(x - h) y − k = ± a b ( x − h ) Eccentricity
e = c a , e > 1 e = \frac{c}{a},\ e > 1 e = a c , e > 1 Hyperbola (Vertical Transverse Axis) Standard Form
( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 a 2 ( y − k ) 2 − b 2 ( x − h ) 2 = 1 Vertices
( h , k ± a ) (h,\ k \pm a) ( h , k ± a ) Foci
( h , k ± c ) , c = a 2 + b 2 (h,\ k \pm c),\ c = \sqrt{a^2 + b^2} ( h , k ± c ) , c = a 2 + b 2 Asymptotes
y − k = ± a b ( x − h ) y - k = \pm\tfrac{a}{b}(x - h) y − k = ± b a ( x − h ) General Conic Discriminant General Form
A x 2 + B x y + C y 2 + D x + E y + F = 0 A x^2 + B xy + C y^2 + D x + E y + F = 0 A x 2 + B x y + C y 2 + D x + E y + F = 0 Discriminant
Δ = B 2 − 4 A C \Delta = B^2 - 4AC Δ = B 2 − 4 A C 