Conics in polar form
Problem 8.463 · medium
For the conic \( \displaystyle r = \frac{10}{4 - \sin{\left(\theta \right)}} \), 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).
- \[ \frac{5}{2 \left(1 - \frac{\sin{\left(\theta \right)}}{4}\right)} = \frac{10}{4 - \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-1}\right|}{4} = \frac{1}{4} \]e = 1/4: < 1, an ellipse.✓ Proved
- \[ \frac{10}{4 \cdot 1 \cdot \frac{1}{4}} = 10 \]ed = 5/2, so d = 10.✓ Proved
Answer \( e = \frac{1}{4},\ \text{ellipse},\ \text{directrix } y = -10 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ) when the given equation involves sinθ, and it does not justify the division step. This misleads a student into using the wrong form and sign.qwen3.6:27b-mlx: fail (error) — The solution correctly identifies the eccentricity and the conic type, but fails to determine the correct directrix. The term -sin(theta) in the denominator corresponds to a directrix of the form y = -d. Since d = 10, the directrix is y = -10. However, the solution's derivation of d is ambiguous and does not explicitly link the sign of the trigonometric term to the position of the directrix, potentially leading to confusion about whether it is y = 10 or y = -10. More critically, standard form is r = ed / (1 - e sin theta) for directrix y = -d. Here we have r = (10/4) / (1 - (1/4)sin theta). So ed = 10/4 and e = 1/4, implying d = 10. The directrix is y = -10. The stated answer is correct, but the solution steps are insufficiently rigorous in establishing the sign of the directrix from the negative sine term. Actually, looking closer, step 4 calculates d=10 but doesn't state the directrix equation. The final answer provided in the problem statement is y=-10. The solution text ends at step 4. It does not explicitly state the directrix equation in the steps provided, only calculating d. If the 'Stated answer' is what the student wrote, and the solution is the work, the work is incomplete because it doesn't conclude with the directrix equation. However, usually these reviews check if the work supports the answer. The work calculates d=10. It misses the crucial step of assigning the sign/location based on the -sin(theta) term. A student might think directrix is y=10. Thus it is misleading/incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution correctly identifies the eccentricity and the conic type, but fails to determine the correct directrix. The term -sin(theta) in the denominator corresponds to a directrix of the form y = -d. Since d = 10, the directrix is y = -10. However, the solution's derivation of d is ambiguous and does not explicitly link the sign of the trigonometric term to the position of the directrix, potentially leading to confusion about whether it is y = 10 or y = -10. More critically, standard form is r = ed / (1 - e sin theta) for directrix y = -d. Here we have r = (10/4) / (1 - (1/4)sin theta). So ed = 10/4 and e = 1/4, implying d = 10. The directrix is y = -10. The stated answer is correct, but the solution steps are insufficiently rigorous in establishing the sign of the directrix from the negative sine term. Actually, looking closer, step 4 calculates d=10 but doesn't state the directrix equation. The final answer provided in the problem statement is y=-10. The solution text ends at step 4. It does not explicitly state the directrix equation in the steps provided, only calculating d. If the 'Stated answer' is what the student wrote, and the solution is the work, the work is incomplete because it doesn't conclude with the directrix equation. However, usually these reviews check if the work supports the answer. The work calculates d=10. It misses the crucial step of assigning the sign/location based on the -sin(theta) term. A student might think directrix is y=10. Thus it is misleading/incomplete.gpt-oss:20b: fail (misleading) 2026-10-19 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ) when the given equation involves sinθ, and it does not justify the division step. This misleads a student into using the wrong form and sign.qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the directrix as y = -10. For the form r = ed / (1 - e sin theta), the directrix is y = -d. Since d = 10, the digpt-oss:20b: fail (error) 2026-10-16 — The solution misidentifies the standard polar form (it uses cos θ instead of sin θ) and consequently derives the wrong directrix sign. The correct directrix for r=10/(4−sinθ) is y=10, not y=−10.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the directrix as y = -10. For the form r = ed/(1 - e sin theta), the directrix is y = -d. Since d = 10, the directrix is y = -10. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin theta). Here we have r = (5/2) / (1 - (1/4) sin theta). So ed = 5/2 and e = 1/4. Thus d = (5/2) / (1/4) = 10. The term is -e sin theta, which corresponds to a directrix of y = -d. So y = -10. The stated answer says y = -10. The solution says d=10. It does not explicitly state the equation of the directrix in the steps, but the final answer matches. However, step 1 says 'Divide top and bottom... to get r = ed/(1 ± e cos theta)'. The problem involves sin, not cos. This is a misleading statement about the form. Also, step 4 calculates d=10 but doesn't explicitly link it to the directrix equation y=-10 in the text, though the final answer has it. The main error is in sentence 1 claiming the target form has cos theta when the problem has sin theta.gpt-oss:20b: fail (misleading) 2026-10-15 — Sentence 1 incorrectly states the standard form uses cos θ, while the given equation involves sin θ. The rest of the reasoning is correct.
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 2026-10-19 with SymPy 1.14.0.