How do you prove that the no-r polar equation #a cot theta + b tan theta = c# represents a pair of straight lines?

Question

1188 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-08-12T03:42:18

See explanation.

In the Cartesian frame, this equation becomes

#a (x/y) + b (y/x) = c#, giving

#a X^2 + b y^2 - c x y = 03

that represents O, when c = 0, a = b,

just one straight line, when #c^2 = 4 a b# and, otherwise,

a pair of real ( or imaginary, when ^ c^2 - 4 a b < 0# ) straights

lines, through the pole, r = 0.

Example.

a - 1, b = 3, c = 5. The equation is #cot theta + 2 tan theta = 4#

See graph, using the Cartesian form #x^2 +3 y^2 - 5 x y = 0#.
graph{x^2+3y^2-5xy=0}