Transcript Document

Example: Find the rectangular equation of the following polar
equations.
(a) 𝑟 = 2 sin  + 6cos 
Multiply both sides of the equation by r
𝑟2 = 2 𝑟sin  + 6𝑟cos  which gives (𝑥2 + 𝑦2) = 2𝑥 + 6𝑦
Put all terms on the left hand side and complete the square
𝑥2 − 2𝑥 + 𝑦2 − 6𝑦 = 0
(𝑥 − 1)2 + (𝑦 − 3)2 = 10
(b) 𝑟 =
𝑥2 − 2𝑥 + 1 + 𝑦2 − 6𝑦 + 9 = 1 + 9
circle
1
2 sin +6cos 
Multiply across
2 𝑟sin  + 6𝑟cos  = 1 which gives
2𝑦 + 6𝑥 = 1
line
(c) 𝑟 =
1
2+6cos 
Multiply across
2𝑟 + 6𝑟cos  = 1 which gives 2 𝑥2 + 𝑦2 + 6𝑥 = 1
2 𝑥2 + 𝑦2 = (1 - 6x)
then square both sides
4𝑥2 + 4𝑦2 = 1 − 12𝑥 + 36𝑥2
4𝑦2 − 32𝑥2 + 12𝑥 = 1
(d) 𝑟 =
putting all the variables on one side
hyperbola
𝑥2
1
2+2cos 
Multiply across
2𝑟 + 2𝑟cos  = 1 which gives 2 𝑥2 + 𝑦2 + 2𝑥 = 1
2 𝑥2 + 𝑦2 = (1 - 2x)
then square both sides
4𝑥2 + 4𝑦2 = 1 − 4𝑥 + 4𝑥2
4𝑦2 = −4𝑥 + 1
which simplifies to
parabola
(e) 𝑟 =
1
6+2cos 
Multiply across
6𝑟 + 2𝑟cos  = 1 which gives 6 𝑥2 + 𝑦2 + 2𝑥 = 1
6 𝑥2 + 𝑦2 = (1 - 2x)
then square both sides
36𝑥2 + 36𝑦2 = 1 − 4𝑥 + 4𝑥2
putting all the variables on one side
32𝑥2 + 12𝑥 + 36𝑦2 = 1
ellipse