Convert the equation given below, in rectangular coordinates, to the polar form r^2= 5 sec(2θ). x^2-y^2=5

Answer 1

r=(5^2*(1/(cos2θ))^2)^.5

5=(x^4+y^4)^.5

Answer 2

Converting x² - y² = 5 to polar:

Replace x with rcosΘ, and y with rsinΘ:

(rcosΘ)² - (rsinΘ)² = 5

r² (cos²Θ - sin²Θ) = 5

r = √( 5 / (cos²Θ - sin²Θ))

CHECK: First, plot the cartesian equation. At the limits, it looks like y = ±x. If x is <√5, y is undefined. If y = 0, x = ±√5

In polar form, r goes to infinity whenever Θ is 45 + n*90.

Since the absolute value of cos or sin is 1/√2, cos²(45 + n*90) + sin²(45 + n*90) is always 0, so r heads to infinity.

For Θ = 0, we have r = √5 / (1 - 0) = √5

PS: to get it in the requested format, note that sec(2Θ) = 1 / (cos²Θ - sin²Θ)