MATH IV FORMULAS
STRAIGHT LINE
Distance between 2 points
Coordinates of a point inside a segment (midpoint r=1)
𝑑 = √(𝑦2 − 𝑦1 )2 + (𝑥 2 − 𝑥 1 )2
𝑥=
𝑥 1 + 𝑟𝑥 2
1+𝑟
Slope-intercept
Point-slope
Symmetric
𝑦 = 𝑚𝑥 + 𝑏
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥 1 )
𝑥 𝑦
+ =1
𝑎 𝑏
Slope
Angle with the horizontal
Angle between 2 lines
𝑚=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑦=
General
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
Distance from a line to a point
𝑚2 − 𝑚1
𝑡𝑎𝑛𝜃 = |
|
1 + 𝑚1 ∗ 𝑚2
𝜃 = tan−1 𝑚
𝑦1 + 𝑟𝑦2
1 +𝑟
𝑑=
|𝐴𝑥 1 + 𝐵𝑦1 + 𝐶 |
√𝐴2 + 𝐵 2
Area of an obtuse or acute triangle
𝐴 = | 𝑎1 𝑏2 + 𝑎2 𝑐1 + 𝑐2 𝑏1 − 𝑎1 𝑐2 − 𝑎2 𝑏1 − 𝑐1 𝑏2 |/2
Convert from polar to rectangular coordinates
𝑥 = 𝑟𝑐𝑜𝑠𝜃
Convert from rectangular to polar coordinates
𝜃 = tan−1 ( 𝑦⁄𝑥 )
𝑟 = √𝑥 2 + 𝑦 2
𝑦 = 𝑟𝑠𝑖𝑛𝜃
CIRCUMFERENCE
𝑥 2 + 𝑦 2 = 𝑟2
𝐴𝑥 2 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2
ELLIPSE
𝐴𝑥 2 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑥 2 𝑦2
+
=1
𝑎2 𝑏 2
𝑥 2 𝑦2
+
=1
𝑏 2 𝑎2
𝑎2 = 𝑏 2 + 𝑐 2
𝐿𝑅 =
(𝑥 − ℎ)2 (𝑦 − 𝑘)2
+
=1
𝑎2
𝑏2
2𝑏2
𝑎
𝑒=
𝑐
𝑎
(𝑥 − ℎ)2 (𝑦 − 𝑘)2
+
=1
𝑏2
𝑎2
PARABOLA
𝐴𝑥 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑥 2 = ±4𝑝𝑦
𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑦 2 = ±4𝑝𝑥
𝐿𝑅 = 4𝑝
(𝑥 − ℎ)2 = ±4𝑝(𝑦 − 𝑘)
(𝑦 − 𝑘)2 = ±4𝑝(𝑥 − ℎ)
HYPERBOLA
2𝑏2
𝐿𝑅 =
𝑎
𝑐 2 = 𝑎2 + 𝑏 2
𝑒=
𝐴𝑥 2 − 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑥 2 𝑦2
−
=1
𝑎2 𝑏 2
𝑦2 𝑥 2
−
=1
𝑎2 𝑏 2
Translation of the axis
𝑥′ = 𝑥 − ℎ
𝑦′ = 𝑦 − 𝑘
𝑐
𝑎
𝐶𝑦 2 − 𝐴𝑥 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
( 𝑥 − ℎ) 2
𝑎2
(𝑦 − 𝑘)2
−
=1
𝑏2
(𝑦 − 𝑘)2 (𝑥 − ℎ)2
−
=1
𝑎2
𝑏2
Rotation of the axis
𝑥 = 𝑥 ′ 𝑐𝑜𝑠𝜃 − 𝑦′𝑠𝑖𝑛𝜃
𝑦 = 𝑥 ′ 𝑠𝑖𝑛𝜃 + 𝑦′𝑐𝑜𝑠𝜃
𝑡𝑎𝑛 2θ =
𝐵
𝐴−𝐶