Conic Sections

Cheat Sheet · Grade 11 Math
Emil Oliversen
CIRCLE
Standard
(x−h)² + (y−k)² = r²
Centre
(h, k)
Radius
r
General form
x²+y²+Dx+Ey+F = 0
General→Standard: complete the square for x and y
ELLIPSE
Horizontal
x²/a² + y²/b² = 1 (a > b)
Vertical
x²/a² + y²/b² = 1 (b > a)
Major axis
2a (horizontal) or 2b (vertical)
Foci
(±c, 0) or (0, ±c)
Key relation
b² = a² − c² → c² = a²−b²
Foci are always on the major (longer) axis
CONIC PARABOLA
Equidistant from focus and directrix
OpensEquationFocusDirectrix
Up(x−h)²=4c(y−k)(h,k+c)y=k−c
Down(x−h)²=−4c(y−k)(h,k−c)y=k+c
Right(y−k)²=4c(x−h)(h+c,k)x=h−c
Left(y−k)²=−4c(x−h)(h−c,k)x=h+c
Latus rectum
length = 4c
Quadratic link
a = 1/(4c)
COMPLETING THE SQUARE
  • Identify which variable is quadratic
  • Move remaining terms to right side
  • Add (b/2)² to both sides
  • Factor as perfect square
HYPERBOLA
Horizontal
x²/a² − y²/b² = 1
Vertical
y²/b² − x²/a² = 1
Vertices
(±a, 0) or (0, ±b)
Foci
(±c, 0) or (0, ±c)
KEY relation
c² = a² + b² (c > a and b)
Asymptotes
y = ±(b/a)x
GRAPHING HYPERBOLA
  • Draw central box: 2a wide, 2b tall
  • Draw asymptotes through corners
  • Plot foci at ±c from centre
  • Sketch branches touching vertices
ELLIPSE vs HYPERBOLA
Ellipse: c² = a²−b² (subtract)
Hyperbola: c² = a²+b² (add)
Ellipse: + between fractions · Hyperbola: −
COMMON MISTAKES
  • Wrong focal formula for each conic
  • Sign error reading h and k
  • Forgetting to add (b/2)² to BOTH sides
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