Conics in polar form
Problem 8.321 · medium
For the conic \( \displaystyle r = \frac{4}{4 - \cos{\left(\theta \right)}} \), 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).
- \[ \frac{1}{1 - \frac{\cos{\left(\theta \right)}}{4}} = \frac{4}{4 - \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-1}\right|}{4} = \frac{1}{4} \]e = 1/4: < 1, an ellipse.✓ Proved
- \[ \frac{4}{4 \cdot 1 \cdot \frac{1}{4}} = 4 \]ed = 1, so d = 4.✓ Proved
Answer \( e = \frac{1}{4},\ \text{ellipse},\ \text{directrix } x = -4 \)
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 (misleading) — The conclusion that the directrix is x = –4 is incorrect; for r = ed/(1 – e cosθ) with d = 4 the directrix is x = 4.qwen3.6:27b-mlx: fail (error) — The directrix is identified as x = -4, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 4, the directrix is indeed x = -4. However, the solution fails to explicitly state the relationship between the sign in the denominator and the position of the directrix, which is crucial for correctness. More importantly, step 2 shows the conversion to 1 - (1/4)cos(theta), implying the form 1 - e cos(theta). For this form, the directrix is x = -d. The solution calculates d=4 but does not explicitly justify the sign of the directrix equation, leaving a gap in the logic for a student. Wait, let me re-read carefully. The problem asks to find the directrix. The answer says x = -4. The derivation shows d=4. The form is 1 - e cos theta. The directrix for 1 - e cos theta is x = -d. So x = -4 is correct. Is there an error? Let's check the algebra. r = 4/(4 - cos theta) = 1/(1 - 1/4 cos theta). So e = 1/4, ed = 1 => d = 4. Form is 1 - e cos theta. Directrix is x = -d = -4. The solution is actually correct. Why did I think it failed? Let me look closer at the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 4 calculates d=4. It does not state the directrix equation. The stated answer has x=-4. The solution text ends at step 4. The 'Stated answer' is separate. The solution steps do not explicitly conclude 'Therefore the directrix is x = -4'. It just finds d. A student might not know that d=4 implies x=-4 for this specific form. This is a 'misleading' or 'error' in completeness? The prompt asks to review the *solution*. The solution stops at finding d. It does not state the final directrix equation. This is an incomplete solution. However, usually 'find the directrix' implies stating the equation. The solution finds the parameter d. Is that enough? It's ambiguous. Let's look for a harder error. Step 2: (4/4)/((-1/4)*cos(theta) + 1). This is 1 / (1 - 0.25 cos theta). Correct. Step 3: e = 1/4. Correct. Step 4: d = 4. Correct. The missing link is the sign of the directrix. If the student assumes d is just a distance, they might say x=4 or x=-4. The form 1 - e cos theta means the directrix is to the left, x = -d. The solution does not mention this. This is a significant gap. I will mark it as error because the conclusion (directrix equation) does not follow from the premises (value of d) without the unstated theorem about the sign.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The directrix is identified as x = -4, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = -d. Since d = 4, the directrix is indeed x = -4. However, the solution fails to explicitly state the relationship between the sign in the denominator and the position of the directrix, which is crucial for correctness. More importantly, step 2 shows the conversion to 1 - (1/4)cos(theta), implying the form 1 - e cos(theta). For this form, the directrix is x = -d. The solution calculates d=4 but does not explicitly justify the sign of the directrix equation, leaving a gap in the logic for a student. Wait, let me re-read carefully. The problem asks to find the directrix. The answer says x = -4. The derivation shows d=4. The form is 1 - e cos theta. The directrix for 1 - e cos theta is x = -d. So x = -4 is correct. Is there an error? Let's check the algebra. r = 4/(4 - cos theta) = 1/(1 - 1/4 cos theta). So e = 1/4, ed = 1 => d = 4. Form is 1 - e cos theta. Directrix is x = -d = -4. The solution is actually correct. Why did I think it failed? Let me look closer at the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 4 calculates d=4. It does not state the directrix equation. The stated answer has x=-4. The solution text ends at step 4. The 'Stated answer' is separate. The solution steps do not explicitly conclude 'Therefore the directrix is x = -4'. It just finds d. A student might not know that d=4 implies x=-4 for this specific form. This is a 'misleading' or 'error' in completeness? The prompt asks to review the *solution*. The solution stops at finding d. It does not state the final directrix equation. This is an incomplete solution. However, usually 'find the directrix' implies stating the equation. The solution finds the parameter d. Is that enough? It's ambiguous. Let's look for a harder error. Step 2: (4/4)/((-1/4)*cos(theta) + 1). This is 1 / (1 - 0.25 cos theta). Correct. Step 3: e = 1/4. Correct. Step 4: d = 4. Correct. The missing link is the sign of the directrix. If the student assumes d is just a distance, they might say x=4 or x=-4. The form 1 - e cos theta means the directrix is to the left, x = -d. The solution does not mention this. This is a significant gap. I will mark it as error because the conclusion (directrix equation) does not follow from the premises (value of d) without the unstated theorem about the sign.gpt-oss:20b: fail (misleading) 2026-10-17 — The conclusion that the directrix is x = –4 is incorrect; for r = ed/(1 – e cosθ) with d = 4 the directrix is x = 4.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is incorrectly identified as x = -4. Since the term is -cos(theta), the directrix is vertical and located at x = -d, which is x = -4. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here e=1/4, d=4, so directrix is x = -4. The stated answer says x = -4. The solution calculates d=4 but does not explicitly state the sign/direction for the directrix in the text, though the final answer in the problem statement matches. However, looking closely at step 4: it calculates d=4. It does not explicitly link d=4 to x=-4. The prompt asks to review the *solution lines*. The solution lines end at step 4. Step 4 concludes d=4. It does not state the directrix equation. The 'Stated answer' is separate. The solution fails to derive the directrix equation x = -4 from d=4 and the cosine term. It just stops at d=4. This is an incomplete solution, but is it an 'error'? The prompt asks to find eccentricity, identify conic, and find directrix. The solution finds e, identifies ellipse, and finds d. It misses the final step of stating the directrix equation. This is a 'misleading' or 'error' because it doesn't answer the full question. Let's look closer. Step 1 says 'Divide... to get r = ed/(1 +/- e cos theta)'. Step 2 shows the algebra. Step 3 finds e. Step 4 finds d. The solution does not explicitly state the directrix. The user provided 'Stated answer' separately. The solution text itself is incomplete. However, usually, these reviews check for correctness of the steps provided. The steps provided are correct. But does it 'find the directrix'? No, it finds d. Finding d is not finding the directrix. The directrix is a line. So the solution fails to answer the question. Verdict: fail. Severity: error (incomplete solution/wrong conclusion).gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The directrix is identified as x = -4, but the standard form r = ed/(1 - e cos theta) corresponds to a directrix at x = -d. Since d=4, the directrix is indeed x = -4. However, the solution fails to explicitly state the sign convention linking the denominator term (-cos theta) to the position of the directrix (x = -d vs x = d). While the final answer is correct, the reasoning in step 4 is incomplete and potentially misleading regarding how the sign in the denominator determines the directrix location. A more rigorous check: r = 4/(4-cos(theta)) = 1/(1 - 1/4 cos(theta)). Here e=1/4, ed=1 => d=4. The form is 1/(1 - e cos theta), which implies the directrix is x = -d = -4. The solution calculates d=4 but does not justify the negative sign for x. It just states 'directrix x = -4' in the final answer without deriving the sign from the equation structure in the steps provided. Actually, looking closer, step 4 says 'ed=1, so d=4'. It does not derive the sign of the directrix. The final answer has the correct sign, but the steps don't show how the sign was determined. This is a gap in logic. Is it an error? The prompt asks to find the directrix. The solution provides the correct value. But the derivation is missing the crucial link between the minus sign in the denominator and the negative x-coordinate. This is a 'misleading' or 'error' in the explanation. Let's call it an error because the step 'ed=1, so d=4' does not logically lead to 'x=-4' without an unstated theorem about the sign.gpt-oss:20b: pass 2026-10-14
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 2026-10-17 with SymPy 1.14.0.