Conics in polar form
Problem 8.444 · medium
For the conic \( \displaystyle r = \frac{4}{4 - 8 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.
- Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).
- \[ \frac{1}{1 - 2 \cos{\left(\theta \right)}} = \frac{4}{4 - 8 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-8}\right|}{4} = 2 \]e = 2: > 1, a hyperbola.✓ Proved
- \[ \frac{4}{2 \cdot 4} = \frac{1}{2} \]ed = 1, so d = 1/2.✓ Proved
Answer \( e = 2,\ \text{hyperbola},\ \text{directrix } x = -1/2 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | for points on the curve, distance to the focus over distance to the directrix is e |
Reviewers
gpt-oss:20b: fail (misleading) — The solution correctly finds e=2 and d=1/2, but it incorrectly states the directrix as x=-1/2. For the form r=ed/(1‑e cosθ) the directrix is x=+d, so the correct directrix is x=1/2, not -1/2.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] The solution correctly identifies e=2 and d=1/2, but fails to determine the correct sign for the directrix. The standard form derived is r = 1 / (1 - 2 cos(theta)), which corresponds to a directrix of x = -d (since the term is -e cos theta). Thus, the directrix is x = -1/2, but the solution does not explicitly justify the sign, merely calculating d. While the final answer matches, the reasoning in step 4 is incomplete as it ignores the sign convention required to locate the directrix relative to the pole.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The solution correctly identifies e=2 and d=1/2, but fails to determine the correct sign for the directrix. The standard form derived is r = 1 / (1 - 2 cos(theta)), which corresponds to a directrix of x = -d (since the term is -e cos theta). Thus, the directrix is x = -1/2, but the solution does not explicitly justify the sign, merely calculating d. While the final answer matches, the reasoning in step 4 is incomplete as it ignores the sign convention required to locate the directrix relative to the pole.gpt-oss:20b: fail (misleading) 2026-10-18 — The solution correctly finds e=2 and d=1/2, but it incorrectly states the directrix as x=-1/2. For the form r=ed/(1‑e cosθ) the directrix is x=+d, so the correct directrix is x=1/2, not -1/2.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies e=2 and d=1/2, but fails to determine the correct sign for the directrix. The standard form derived is r = 1/(1 - 2cos(theta)), which corresponds to a directrix at x = -d = -1/2. However, the solution simply states d=1/2 without explicitly linking the minus sign in the denominator to the negative x-coordinate, potentially leaving the student confused about why the directrix is x = -1/2 rather than x = 1/2 or just stating the distance.gpt-oss:20b: fail (error) 2026-10-16 — The directrix is incorrectly identified. For r = ed/(1‑e cosθ) the directrix is x = d, so with e=2 and d=½ the directrix is x = ½, not x = –½.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution correctly identifies the eccentricity and conic type but fails to determine the correct sign for the directrix. The standard form requires the denominator to be $1 - e \cos \theta$, implying a directrix at $x = -d$ (since $r = \frac{ed}{1 - e \cos \theta}$ corresponds to a vertical directrix $x = -d$). With $d=1/2$, the directrix is $x = -1/2$. While the final answer happens to be correct, the reasoning in step 4 ($ed=1 \implies d=1/2$) does not explicitly link the negative sign in the denominator ($1 - 2\cos\theta$) to the negative position of the directrix ($x = -1/2$). A student might incorrectly assume the directrix is always $x=d$ or $x=-d$ based on the sign of the cosine term without understanding the geometric derivation. Specifically, the form $r = \frac{ed}{1 - e \cos \theta}$ has directrix $x = -d$. The solution calculates $d=1/2$ but does not explicitly state that the minus sign in the denominator dictates the negative sign for the directrix's x-coordinate, making the derivation of the directrix's location incomplete and potentially misleading if the sign were different.gpt-oss:20b: pass 2026-10-15
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/polar_conics, checked 2026-10-18 with SymPy 1.14.0.