# 10.7.2: Key Equations

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### Key Equations

 Horizontal ellipse, center at origin $x 2 a 2 + y 2 b 2 =1,a>b x 2 a 2 + y 2 b 2 =1,a>b$ Vertical ellipse, center at origin $x 2 b 2 + y 2 a 2 =1,a>b x 2 b 2 + y 2 a 2 =1,a>b$ Horizontal ellipse, center $(h,k) (h,k)$ $( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1,a>b ( x−h ) 2 a 2 + ( y−k ) 2 b 2 =1,a>b$ Vertical ellipse, center $(h,k) (h,k)$ $( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1,a>b ( x−h ) 2 b 2 + ( y−k ) 2 a 2 =1,a>b$
 Hyperbola, center at origin, transverse axis on x-axis $x 2 a 2 − y 2 b 2 =1 x 2 a 2 − y 2 b 2 =1$ Hyperbola, center at origin, transverse axis on y-axis $y 2 a 2 − x 2 b 2 =1 y 2 a 2 − x 2 b 2 =1$ Hyperbola, center at $(h,k), (h,k),$ transverse axis parallel to x-axis $( x−h ) 2 a 2 − ( y−k ) 2 b 2 =1 ( x−h ) 2 a 2 − ( y−k ) 2 b 2 =1$ Hyperbola, center at $(h,k), (h,k),$ transverse axis parallel to y-axis $( y−k ) 2 a 2 − ( x−h ) 2 b 2 =1 ( y−k ) 2 a 2 − ( x−h ) 2 b 2 =1$
 Parabola, vertex at origin, axis of symmetry on x-axis $y 2 =4px y 2 =4px$ Parabola, vertex at origin, axis of symmetry on y-axis $x 2 =4py x 2 =4py$ Parabola, vertex at $(h,k), (h,k),$ axis of symmetry on x-axis $( y−k ) 2 =4p( x−h ) ( y−k ) 2 =4p( x−h )$ Parabola, vertex at $(h,k), (h,k),$ axis of symmetry on y-axis $( x−h ) 2 =4p( y−k ) ( x−h ) 2 =4p( y−k )$
 General Form equation of a conic section $A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0$ Rotation of a conic section $x= x ′ cosθ− y ′ sinθ y= x ′ sinθ+ y ′ cosθ x= x ′ cosθ− y ′ sinθ y= x ′ sinθ+ y ′ cosθ$ Angle of rotation $θ,where cot( 2θ )= A−C B θ,where cot( 2θ )= A−C B$

10.7.2: Key Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.