Identify the conic section given by the polar equation r=41−cos⁡θ and also determine its directrix.

Question: "Identify the conic section given by the polar..."

Kyran Hudson

2021-02-12

Identify the conic section given by the polar equation

r = \frac{4}{1 - \cos θ}

and also determine its directrix.

smallq9

Skilled2021-02-13Added 106 answers

Step 1
We convert the equation into rectangular form first

r = \frac{4}{1 - \cos θ}

r (1 - \cos θ) = 4

r - r \cos θ = 4

\sqrt{x^{2} + y^{2}} - x = 4

\sqrt{x^{2} + y^{2}} = x + 4

x^{2} + y^{2} = {(x + 4)}^{2}

x^{2} + y^{2} = x^{2} + 8 x + 16

y^{2} = 8 x + 16

y^{2} = 8 (x + 2)

This is a parabola.
Step 2
Compare the equation with the standard form

{(y - k)}^{2} = 4 p (x - h)

h = - 2, k = 0

8 p = 2

or, p = 2

Directrix

= x = h - p or x = - 2 - 2 or x = - 4

Answer: Parabola,

x = - 4

Jeffrey Jordon

Expert2021-10-26Added 2605 answers

Answer is given below (on video)

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