Sketch conic r = 4secθ/(2secθ−1)

Multiply numerator and denominator by cosθ:

r = 4 secθ/(2 secθ − 1) = 4/(2 − cosθ) = 2/(1 − (1/2)cosθ)

Compare with r = l/(1 − e cosθ): e = 1/2 < 1, so the conic is an ellipse; l = 2.

Key facts

• ed = 2, so d = 4. With (1 − e cosθ) the directrix is on the left: x = −4.
• The major axis is along the x-axis.
• θ = 0: r = 4/(2 − 1) = 4, vertex (4, 0).
• θ = π: r = 4/3, vertex (−4/3, 0).
• θ = π/2: r = 2, so the latus rectum ends are (0, ±2).
• Major axis 2a = 4 + 4/3 = 16/3, so a = 8/3.
• Centre = (4/3, 0); ae = 4/3 ✓ (the pole is a focus); other focus (8/3, 0).
• b² = a²(1 − e²) = (64/9)(3/4) = 16/3, so b = 4/√3 ≈ 2.31.

           y
    x=-4   |  (0,2)
     |   .-'-.
     |  / F   \
 ----|-(-+--C--)--x
     | (-4/3  4/3  4)
     |  \     /
        '-.-'

Sketch: an ellipse from x = −4/3 to x = 4 on the x-axis, centre (4/3, 0), passing through (0, ±2), with the focus at the origin and the directrix x = −4 on the left.