CONIC SECTIONS
● HOURGLASS (VERTICAL) - Parallel
1. CIRCLE - When the plane is Horizontal to the y-axis
x= h Center (h,k)
y=k Conjugate Axis Horizontal
2 2
Distance Formula: (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Equation Form is
(𝑦−𝑘)
𝑎
2 −
(𝑥−ℎ)
𝑏
2 =1
(𝑥2−𝑥1) (𝑦2−𝑦1) Vertices (h, k ± a )
Midpoint Formula: 2
, 2
Co-Vertices (h ± b, k)
Slope-Intercept form: y = mx+b ; m is the
negative reciprocal of the slope. Foci (h, k ± c )
𝑎
𝑦2−𝑦1 Asymptotes 𝑦 = ± 𝑏
(𝑥 − ℎ) + 𝑘
Slope of the Line: 𝑀 = 𝑥2−𝑥1 2
𝑏
Lactus Rectum 𝑎
2 2 2
Standard equation:(𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑟
2 2
Transverse Length = 2a
General form: 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0,
𝑤ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0. 3. PARABOLA-
Solve the radius in tangent x-axis or y-axis by Vertex : origin V (0, 0) If the parabola
counting based on the given axis. opens upward, the vertex is the
Completing the square. lowest point. If the Parabola opens
(Note: both sides should be equally added) downward, The vertex is the
In an equation, if A is not equal to 1, make Highest point.
sure to factor out leaving the standard Directrix : the line y = −c or y = c
equation reduced. When adding to the other The directrix is c units below or above
side,don’t forget to multiply the used factor. the vertex.
System of linear equations by using Focus: F(0, c) or F(0, −c) The focus is
substitution to find out the value of x and y. c units above or below the Vertex. Any
When it has a diameter with endpoints use point on the parabola has the same
the Distance Formula. distance from the focus as it has from
the directrix.
2. HYPERBOLA - when the plane (not necessarily
Axis of symmetry: x = 0 (the y-axis)
vertical) intersect both cones to form two unbounded
This line divides the parabola into two
curves (each called a branch of the hyper-bola)
parts which are mirror images of each
other.
● BUTTERFLY (HORIZONTAL) - parallel to
the x-axis
FORMULA VERTEX AT ORIGIN
2
Center (h,k) 𝑥 = 4𝑎𝑦 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝𝑤𝑎𝑟𝑑
2 2
(𝑥−ℎ) (𝑦−𝑘) 2
Equation form is 2 + 2 = 1 𝑥 = − 4𝑎𝑦 𝑜𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑
𝑎 𝑏
2
Transverse Axis = Horizontal 𝑦 = − 4𝑎𝑥 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡
2
Transverse Length = 2a 𝑦 = 4𝑎𝑥 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡
Conjugate Axis = Vertical
Conjugate Length = 2b FORMULA VERTEX AT (H,K)
2
vertex (ℎ ± 𝑎, 𝑘) (𝑥 − ℎ) = 4𝑎 (𝑦 − 𝑘) 𝑢𝑝𝑤𝑎𝑟𝑑
2
Co-vertex (ℎ, 𝑘 ± 𝑏) (𝑥 − ℎ) =− 4𝑎 (𝑦 − 𝑘) 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑
Foci (ℎ ± 𝑐, 𝑘) (𝑦 − 𝑘)
2
=− 4𝑎 (𝑥 − ℎ) 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓
𝑏
Asymptotes 𝑦 = ± (𝑥 − ℎ) + 𝑘 2
𝑎 (𝑦 − 𝑘) = 4𝑎 (𝑥 − ℎ) 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ
2 2 2
Pythagorean theorem: 𝑎 + 𝑏 = 𝑐
● HOURGLASS (VERTICAL) - Parallel
1. CIRCLE - When the plane is Horizontal to the y-axis
x= h Center (h,k)
y=k Conjugate Axis Horizontal
2 2
Distance Formula: (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Equation Form is
(𝑦−𝑘)
𝑎
2 −
(𝑥−ℎ)
𝑏
2 =1
(𝑥2−𝑥1) (𝑦2−𝑦1) Vertices (h, k ± a )
Midpoint Formula: 2
, 2
Co-Vertices (h ± b, k)
Slope-Intercept form: y = mx+b ; m is the
negative reciprocal of the slope. Foci (h, k ± c )
𝑎
𝑦2−𝑦1 Asymptotes 𝑦 = ± 𝑏
(𝑥 − ℎ) + 𝑘
Slope of the Line: 𝑀 = 𝑥2−𝑥1 2
𝑏
Lactus Rectum 𝑎
2 2 2
Standard equation:(𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑟
2 2
Transverse Length = 2a
General form: 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0,
𝑤ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0. 3. PARABOLA-
Solve the radius in tangent x-axis or y-axis by Vertex : origin V (0, 0) If the parabola
counting based on the given axis. opens upward, the vertex is the
Completing the square. lowest point. If the Parabola opens
(Note: both sides should be equally added) downward, The vertex is the
In an equation, if A is not equal to 1, make Highest point.
sure to factor out leaving the standard Directrix : the line y = −c or y = c
equation reduced. When adding to the other The directrix is c units below or above
side,don’t forget to multiply the used factor. the vertex.
System of linear equations by using Focus: F(0, c) or F(0, −c) The focus is
substitution to find out the value of x and y. c units above or below the Vertex. Any
When it has a diameter with endpoints use point on the parabola has the same
the Distance Formula. distance from the focus as it has from
the directrix.
2. HYPERBOLA - when the plane (not necessarily
Axis of symmetry: x = 0 (the y-axis)
vertical) intersect both cones to form two unbounded
This line divides the parabola into two
curves (each called a branch of the hyper-bola)
parts which are mirror images of each
other.
● BUTTERFLY (HORIZONTAL) - parallel to
the x-axis
FORMULA VERTEX AT ORIGIN
2
Center (h,k) 𝑥 = 4𝑎𝑦 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝𝑤𝑎𝑟𝑑
2 2
(𝑥−ℎ) (𝑦−𝑘) 2
Equation form is 2 + 2 = 1 𝑥 = − 4𝑎𝑦 𝑜𝑝𝑒𝑛𝑠 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑
𝑎 𝑏
2
Transverse Axis = Horizontal 𝑦 = − 4𝑎𝑥 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡
2
Transverse Length = 2a 𝑦 = 4𝑎𝑥 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡
Conjugate Axis = Vertical
Conjugate Length = 2b FORMULA VERTEX AT (H,K)
2
vertex (ℎ ± 𝑎, 𝑘) (𝑥 − ℎ) = 4𝑎 (𝑦 − 𝑘) 𝑢𝑝𝑤𝑎𝑟𝑑
2
Co-vertex (ℎ, 𝑘 ± 𝑏) (𝑥 − ℎ) =− 4𝑎 (𝑦 − 𝑘) 𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑
Foci (ℎ ± 𝑐, 𝑘) (𝑦 − 𝑘)
2
=− 4𝑎 (𝑥 − ℎ) 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑒𝑓
𝑏
Asymptotes 𝑦 = ± (𝑥 − ℎ) + 𝑘 2
𝑎 (𝑦 − 𝑘) = 4𝑎 (𝑥 − ℎ) 𝑜𝑝𝑒𝑛𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ
2 2 2
Pythagorean theorem: 𝑎 + 𝑏 = 𝑐
