Question

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

Answer 1

See explanation.

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}