Conics in polar form
Problem 8.433 · medium
For the conic \( \displaystyle r = \frac{12}{4 - 5 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - \frac{5 \sin{\left(\theta \right)}}{4}} = \frac{12}{4 - 5 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-5}\right|}{4} = \frac{5}{4} \]e = 5/4: > 1, a hyperbola.✓ Proved
- \[ \frac{12}{4 \cdot \frac{5}{4}} = \frac{12}{5} \]ed = 3, so d = 12/5.✓ Proved
Answer \( e = \frac{5}{4},\ \text{hyperbola},\ \text{directrix } y = -12/5 \)
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 solution misidentifies the standard form: for a denominator involving sinθ the correct form is r = ed/(1 ± e sinθ), not cosθ. Consequently the directrix sign is wrong; with r = 3/(1‑(5/4)sinθ) the directrix is y = +12/5, not y = –12/5. The setup and conclusion are therefore incorrect.qwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as y = -12/5. For the form r = ed/(1 - e sin(theta)), the directrix is y = -d. Since d = 12/5, the directrix is y = -12/5. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin(theta)). Here we have r = (15/4) / (1 - (5/4)sin(theta))? No. Let's look at step 2. r = 3 / (1 - 1.25 sin(theta)). So ed = 3, e = 1.25. d = 3/1.25 = 2.4 = 12/5. The form is 1 - e sin(theta), which corresponds to a directrix at y = -d. So y = -12/5. The stated answer says y = -12/5. The solution says d = 12/5. It does not explicitly state the equation of the directrix in the sentences, but the stated answer matches. However, step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a misleading statement about the form. Also, step 4 calculates d but doesn't explicitly link it to the directrix equation y = -d. The prompt asks to find the directrix. The solution calculates d. The stated answer provides the directrix. The review is of the solution lines. Line 1 is misleading/wrong because it mentions cos instead of sin. Line 4 calculates d but doesn't conclude the directrix equation. The verdict should be fail because the setup in line 1 is incorrect (mentions cos for a sin problem) and the solution doesn't explicitly state the directrix equation, only d.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The directrix is incorrectly identified as y = -12/5. For the form r = ed/(1 - e sin(theta)), the directrix is y = -d. Since d = 12/5, the directrix is y = -12/5. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin(theta)). Here we have r = (15/4) / (1 - (5/4)sin(theta))? No. Let's look at step 2. r = 3 / (1 - 1.25 sin(theta)). So ed = 3, e = 1.25. d = 3/1.25 = 2.4 = 12/5. The form is 1 - e sin(theta), which corresponds to a directrix at y = -d. So y = -12/5. The stated answer says y = -12/5. The solution says d = 12/5. It does not explicitly state the equation of the directrix in the sentences, but the stated answer matches. However, step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a misleading statement about the form. Also, step 4 calculates d but doesn't explicitly link it to the directrix equation y = -d. The prompt asks to find the directrix. The solution calculates d. The stated answer provides the directrix. The review is of the solution lines. Line 1 is misleading/wrong because it mentions cos instead of sin. Line 4 calculates d but doesn't conclude the directrix equation. The verdict should be fail because the setup in line 1 is incorrect (mentions cos for a sin problem) and the solution doesn't explicitly state the directrix equation, only d.gpt-oss:20b: fail (error) 2026-10-18 — The solution misidentifies the standard form: for a denominator involving sinθ the correct form is r = ed/(1 ± e sinθ), not cosθ. Consequently the directrix sign is wrong; with r = 3/(1‑(5/4)sinθ) the directrix is y = +12/5, not y = –12/5. The setup and conclusion are therefore incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The directrix is incorrect. For r = ed/(1 - e sin theta), the directrix is y = -d. Here d = 12/5, so the directrix is y = -12/5. Wait, the stated answer says y = -12/5. Let me re-read carefully. Standard form: r = ed / (1 - e sin theta). This corresponds to a directrix at y = -d. From step 2: r = 3 / (1 - (5/4) sin theta). So ed = 3, e = 5/4. d = 3 / (5/4) = 12/5. Since the term is -e sin theta, the directrix is y = -d = -12/5. The stated answer is y = -12/5. The solution steps calculate d = 12/5 but do not explicitly state the sign of the directrix in the text, though the final answer provided in the problem statement matches the correct derivation. However, the solution text itself ends at step 4 calculating d. It does not explicitly conclude the directrix equation. The prompt asks to review the *solution*. The solution stops at calculating d. It does not state the final directrix equation. The 'Stated answer' is separate. The solution provided is incomplete as it doesn't state the directrix. But usually, these reviews check if the steps lead to the answer. Step 4 calculates d. It misses the step connecting d to the directrix equation y = -d. Is this an error? It's an incomplete solution. But is it 'wrong'? Let's look closer. Step 1 says 'Divide ... to get r = ed/(1 ± e cos theta)'. It mentions cos, but the problem has sin. This is a minor style/inaccuracy in the sentence description, but the math follows sin. The main issue: The solution calculates d = 12/5. It does not state the directrix is y = -12/5. The 'Stated answer' has it. The solution text provided is just 4 steps. Step 4 ends with 'd = 12/5'. It fails to identify the directrix equation, which is part of the problem request ('find the directrix'). Therefore, the solution is incomplete. However, often 'find the directrix' implies finding d and its position. The sign is crucial. The solution does not determine the sign. It just calculates the magnitude d. Without stating the sign convention (that -sin implies y = -d), the student might think the directrix is y = 12/5. Thus, the solution is misleading/incomplete regarding the final answer. Actually, looking at the provided 'Stated answer', it is correct. The solution steps are correct up to finding d. The missing link is the final conclusion. Is that an 'error'? It's a missing step. But more importantly, Step 1 says 'get r = ed/(1 ± e cos theta)'. The problem involves sine. This is a factual error in the sentence describing the method. It should say sin. This is a 'misleading' or 'error' in the instructional text. Let's check the directrix sign again. r = ed / (1 - e sin theta). Directrix is y = -d. d = 12/5. Directrix y = -12/5. Correct. The sentence in Step 1 is wrong: it says 'cos' but the problem is 'sin'. This is a clear error in the explanation.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly states the standard form uses γω instead of σ and omits the sign convention for the “1±e” term. It also does not explain why the directrix is y = –12/5. These inaccuracies could mislead a student.qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution correctly identifies the eccentricity and the type of conic, but it fails to determine the correct sign for the directrix. The term -5 sin(theta) indicates the directrix is below the pole (y = -d), so the directrix should be y = -12/5, not y = 12/5. The calculation of d=12/5 is correct, but the final statement of the directrix equation lacks the negative sign required by the sine term's sign.gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cos θ) and claims a division by the constant in the denominator, but the given equation involves sin θ, not cos θ. This misleads a student about the correct polar form and the sign convention.
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.