Sketch conic r = 12/(6+2sinθ)

Divide by 6 to get the form r = l/(1 + e sinθ):

r = 12/(6 + 2sinθ) = 2/(1 + (1/3) sinθ)

So e = 1/3 < 1: an ellipse, with the pole at a focus and l = 2.

Key facts

• ed = 2, so d = 6: the directrix is the horizontal line y = 6.
• The major axis lies along the y-axis (sinθ term).
• θ = π/2: r = 12/8 = 3/2, vertex (0, 3/2).
• θ = 3π/2: r = 12/4 = 3, vertex (0, −3).
• θ = 0 or π: r = 2, so the latus rectum ends are (±2, 0).
• Major axis 2a = 3/2 + 3 = 9/2, so a = 9/4. Centre (0, −3/4); ae = 3/4 (the pole is 3/4 from the centre ✓).
• b² = a²(1 − e²) = (81/16)(8/9) = 9/2, so b = 3/√2 ≈ 2.12.
• Other focus: (0, −3/2).

         y
         |      y = 6
  - - - - - - - - - - - -
         | (0,1.5)
       .-'-.
      /  F  \ (2,0)
 ----(---+---)----x
      \  C  /
       '-.-'
         | (0,-3)

The ellipse is long vertically, from y = −3 to y = 3/2, with a focus at the origin and the directrix y = 6 above it.