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Key Equations

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