Conics in polar form
Problem 8.441 · medium
For the conic \( \displaystyle r = \frac{8}{2 - 2 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{4}{1 - \cos{\left(\theta \right)}} = \frac{8}{2 - 2 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-2}\right|}{2} = 1 \]e = 1: = 1, a parabola.✓ Proved
- \[ \frac{8}{1 \cdot 2} = 4 \]ed = 4, so d = 4.✓ Proved
Answer \( e = 1,\ \text{parabola},\ \text{directrix } x = -4 \)
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 (error) — The directrix is incorrectly identified. For r = 4/(1‑cosθ) the directrix is x = +4, not x = –4.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies the eccentricity and distance $d$, but fails to determine the correct sign for the directrix. The standard form $r = \frac{ed}{1 - e \cos \theta}$ corresponds to a directrix at $x = -d$. Since $d=4$, the directrix is $x = -4$. However, the solution merely states $d=4$ without explicitly linking the minus sign in the denominator to the negative x-coordinate, which is a critical step in identifying the directrix's location. While the final answer is correct, the reasoning skips the crucial sign analysis required to distinguish between $x=4$ and $x=-4$.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies the eccentricity and distance $d$, but fails to determine the correct sign for the directrix. The standard form $r = \frac{ed}{1 - e \cos \theta}$ corresponds to a directrix at $x = -d$. Since $d=4$, the directrix is $x = -4$. However, the solution merely states $d=4$ without explicitly linking the minus sign in the denominator to the negative x-coordinate, which is a critical step in identifying the directrix's location. While the final answer is correct, the reasoning skips the crucial sign analysis required to distinguish between $x=4$ and $x=-4$.gpt-oss:20b: fail (error) 2026-10-18 — The directrix is incorrectly identified. For r = 4/(1‑cosθ) the directrix is x = +4, not x = –4.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution correctly identifies e=1 and d=4, but fails to determine the correct sign for the directrix. The standard form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d. Since the denominator is 1 - cos(theta), the directrix is x = -4. However, the solution simply states 'd=4' without explicitly linking the negative sign in the denominator to the position of the directrix (x = -d vs x = +d), which is a critical step in identifying the directrix's location. While the final answer x=-4 is correct, the reasoning in step 4 is incomplete and potentially misleading if a student assumes d always implies x=d. A more rigorous check: The form is r = ed/(1 - e cos theta). Here 1 - 1*cos(theta) matches. This form corresponds to directrix x = -d. So x = -4. The solution says 'd=4' but doesn't explicitly state the rule for the sign. It's a 'misleading' or 'error' in explanation. Let's look closer. Step 4 says 'ed=4, so d=4'. It does not conclude 'directrix x = -4'. The stated answer has x=-4. The solution text provided ends at step 4. Does step 4 imply the directrix? No. It just finds d. The connection between the sign in the denominator and the sign of the directrix is missing. This is a logical gap.gpt-oss:20b: fail (style) 2026-10-16 — The solution correctly computes e and d but never states the directrix x = –4, so it is incomplete for the problem’s requirement.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the directrix as x = -4. The standard form derived is r = 4 / (1 - cos(theta)), which corresponds to a directrix of x = -d, i.e., x = -4. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos theta) has directrix x = -d. Here e=1, d=4, so directrix is x = -4. The stated answer says directrix x = -4. The solution steps are correct. Why did I think it was wrong? Let me check the sign convention carefully. r = ed / (1 - e cos theta). If the denominator is 1 - cos theta, the focus is at the origin, and the directrix is x = -d. So x = -4 is correct. The solution is actually correct. Let me re-read the prompt's 'Stated answer'. Stated answer: e = 1, parabola, directrix x = -4. Solution derives e=1, d=4. It does not explicitly state the directrix equation in the text, but step 4 calculates d=4. The prompt asks to review the solution. The solution stops at calculating d. It does not explicitly write 'directrix x = -4'. However, usually, finding d is sufficient if the form is identified. But wait, does the solution explicitly conclude the directrix? No. It just calculates d. Is that an error? The problem asks to 'find the directrix'. The solution finds d but doesn't state the equation of the directrix. This is a 'misleading' or 'error' in completeness? Or is it just implicit? Let's look closer. Step 1 says 'get r = ed/(1 +/- e cos theta)'. Step 2 shows the algebra. Step 3 finds e. Step 4 finds d. It never states the directrix is x = -4. It just says 'd=4'. A student might not know that d=4 implies x=-4 for the minus sign. This is a gap. However, is it 'wrong'? It's incomplete. But looking at the 'Stated answer', it matches. The solution *derives* the components. Is the derivation correct? Yes. Is the conclusion missing? Yes. But usually, these reviews focus on correctness of the math shown. The math shown is correct. The setup is correct. The conclusion 'd=4' is correct. The mapping from d to the line equation is standard knowledge. I will mark it as pass because the mathematical content is correct, even if the final sentence is implicit.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.