Analyze the equation and graph r = 1/1 + cos theta?

Question

Please help. step by step process of finding the vertex, directrix, other necessary points to plot.

Answer 1

The Function is r =  1/(1+cosθ)

We know that in polar coordinates x² + y² = r² , x = rcosθ and y = rsinθ.

cosθ = x/r

Then the Function written as r = 1/(1+x/r)

Multiply with (1+x/r) on both sides.

r(1+x/r) = 1

r + x = 1

r = (1 - x)

Squaring on both sides

r² = (1 - x)²

x² + y² = 1 + x² - 2x

y² = 1 - 2x

y² = 2(x - (-1/2))

This Equation is in the form of (y-k)² = 4p(x-h) which represents a parabola equation.

Parabola y² = -2(x - 1/2) open towards right side.

Vertex (h, k) = (-1/2,0)

To find the focus and directrix of this parabola:

It has the form y = ax² + bx +c.

x = (1/2)y² - (1/2)

Therefore a  = 1/2, b = 0 and c = -1/2.

distance from the focus to the vertex p = 1/4a

p = 1/a(1/2) = 1/2

Focus = (h+p, k) and Directrix is x =h - p

Focus = (0,0)  and Directrix x = -1

Graph

Therefore the parabola Equation is y² = 2(x - (-1/2)) where Vertex (-1/2,0), Focus is (0,0) and Directrix x = -1.