Sketch conic r = 10/(3+2cosθ)

Write the equation in the standard polar form r = l/(1 + e cosθ), with the pole at a focus.

r = 10/(3 + 2cosθ) = (10/3) / (1 + (2/3)cosθ)

So e = 2/3 < 1, and the conic is an ellipse, with semi-latus rectum l = 10/3.

Key facts

• ed = l, so d = (10/3)/(2/3) = 5: directrix x = 5 (to the right of the pole).
• The major axis lies along the x-axis (cosθ term).
• θ = 0: r = 10/5 = 2, vertex (2, 0).
• θ = π: r = 10/1 = 10, vertex (−10, 0).
• θ = π/2: r = 10/3, so the ends of the latus rectum are (0, ±10/3).

Size of the ellipse

• Major axis 2a = 2 + 10 = 12, so a = 6.
• Centre = mid-point of the vertices = (−4, 0). Focal distance ae = 6 · 2/3 = 4, which matches: the pole (focus) is 4 units from the centre.
• b² = a²(1 − e²) = 36 · 5/9 = 20, so b = 2√5 ≈ 4.47.
• Other focus: (−8, 0).

              y
              |        x = 5
          ____|____    |
        .'    |    '.  |
  -----(--F2--C--F1)--------- x
     -10 -8  -4  0 2
        '.____|____.'
              |

Sketch: a horizontal ellipse with vertices (−10, 0) and (2, 0), centre C(−4, 0), top and bottom at (−4, ±2√5), one focus F1 at the origin, the other F2 at (−8, 0), and the directrix x = 5 as a vertical line to the right of the curve.