Conics in polar form
Problem 8.462 · medium
For the conic \( \displaystyle r = \frac{9}{3 - 8 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{3}{1 - \frac{8 \sin{\left(\theta \right)}}{3}} = \frac{9}{3 - 8 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-8}\right|}{3} = \frac{8}{3} \]e = 8/3: > 1, a hyperbola.✓ Proved
- \[ \frac{9}{3 \cdot \frac{8}{3}} = \frac{9}{8} \]ed = 3, so d = 9/8.✓ Proved
Answer \( e = \frac{8}{3},\ \text{hyperbola},\ \text{directrix } y = -9/8 \)
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: passqwen3.6:27b-mlx: fail (error) — The directrix is incorrect. For the form r = ed / (1 - e sin theta), the directrix is y = -d. Here d = 9/8, so the directrix is y = -9/8. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin theta). The given equation is r = 9 / (3 - 8 sin theta) = 3 / (1 - (8/3) sin theta). So ed = 3 and e = 8/3. Thus d = 3 / (8/3) = 9/8. Since the term is -e sin theta, the directrix is y = -d = -9/8. The stated answer says y = -9/8. The solution says d = 9/8 but doesn't explicitly state the sign of the directrix equation, just 'd'. However, usually 'directrix' implies the line equation. Let's look closer. Step 4 calculates d. It does not state the final directrix equation. The stated answer has y = -9/8. Is the solution's derivation of d correct? Yes. Does it conclude the directrix? No, it just finds d. But the prompt asks to review the solution lines. Line 4 says 'ed = 3, so d = 9/8'. This is correct algebra. The missing link is connecting d to the line equation y = -d. Is this an error or just incomplete? The problem asks to 'find the directrix'. The solution stops at finding d. A student might think d is the directrix. This is misleading/incomplete. However, looking at the stated answer, it matches. Let's look at the sign again. r = ed / (1 - e sin theta). Directrix is y = -d. r = ed / (1 + e sin theta). Directrix is y = d. Here we have 1 - (8/3) sin theta. So y = -9/8. The solution finds d=9/8. It does not explicitly write y = -9/8. This is a gap. But is it an 'error' in the provided lines? The lines themselves are algebraically correct. The sentence in line 1 says 'get r = ed/(1 +/- e cos theta)'. It mentions cos, but the problem has sin. This is a slight inaccuracy in the general formula description, but the subsequent steps use sin. Line 1 is slightly misleading by only mentioning cos. Line 4 is correct algebra. The final answer is correct. The main issue is Line 1 mentioning cos instead of sin, which is a style/minor error, but the bigger issue is that the solution doesn't explicitly state the directrix equation, only d. If I must choose, the omission of the final step (converting d to the line equation) makes the solution incomplete, but the lines provided are technically correct statements. However, Line 1 says 'Divide... to get r = ed/(1 +/- e cos theta)'. This is false for this problem which uses sin. This is a misleading statement about the standard form. It should be 'cos or sin'. This is a 'misleading' severity.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The directrix is incorrect. For the form r = ed / (1 - e sin theta), the directrix is y = -d. Here d = 9/8, so the directrix is y = -9/8. Wait, let me re-evaluate. The standard form is r = ed / (1 - e sin theta). The given equation is r = 9 / (3 - 8 sin theta) = 3 / (1 - (8/3) sin theta). So ed = 3 and e = 8/3. Thus d = 3 / (8/3) = 9/8. Since the term is -e sin theta, the directrix is y = -d = -9/8. The stated answer says y = -9/8. The solution says d = 9/8 but doesn't explicitly state the sign of the directrix equation, just 'd'. However, usually 'directrix' implies the line equation. Let's look closer. Step 4 calculates d. It does not state the final directrix equation. The stated answer has y = -9/8. Is the solution's derivation of d correct? Yes. Does it conclude the directrix? No, it just finds d. But the prompt asks to review the solution lines. Line 4 says 'ed = 3, so d = 9/8'. This is correct algebra. The missing link is connecting d to the line equation y = -d. Is this an error or just incomplete? The problem asks to 'find the directrix'. The solution stops at finding d. A student might think d is the directrix. This is misleading/incomplete. However, looking at the stated answer, it matches. Let's look at the sign again. r = ed / (1 - e sin theta). Directrix is y = -d. r = ed / (1 + e sin theta). Directrix is y = d. Here we have 1 - (8/3) sin theta. So y = -9/8. The solution finds d=9/8. It does not explicitly write y = -9/8. This is a gap. But is it an 'error' in the provided lines? The lines themselves are algebraically correct. The sentence in line 1 says 'get r = ed/(1 +/- e cos theta)'. It mentions cos, but the problem has sin. This is a slight inaccuracy in the general formula description, but the subsequent steps use sin. Line 1 is slightly misleading by only mentioning cos. Line 4 is correct algebra. The final answer is correct. The main issue is Line 1 mentioning cos instead of sin, which is a style/minor error, but the bigger issue is that the solution doesn't explicitly state the directrix equation, only d. If I must choose, the omission of the final step (converting d to the line equation) makes the solution incomplete, but the lines provided are technically correct statements. However, Line 1 says 'Divide... to get r = ed/(1 +/- e cos theta)'. This is false for this problem which uses sin. This is a misleading statement about the standard form. It should be 'cos or sin'. This is a 'misleading' severity.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The directrix is incorrectly identified as y = -9/8. Since the equation involves sin(theta) and the denominator is 1 - e sin(theta) (after dividing by 3), the directrix is horizontal and below the pole, corresponding to y = -d, which is y = -9/8. Wait, let me re-evaluate. Standard form r = ed / (1 - e sin theta) corresponds to directrix y = -d. Here we have r = 3 / (1 - (8/3) sin theta). So ed = 3, e = 8/3, d = 9/8. The form is 1 - e sin theta, which implies the directrix is y = -d = -9/8. The stated answer says y = -9/8. The solution calculates d = 9/8 but does not explicitly state the sign/direction logic for the directrix location, merely calculating d. However, looking closely at step 1, it says 'Divide top and bottom... to get r = ed/(1 ± e cos theta)'. It mentions cos theta but the problem has sin theta. This is a misleading statement about the target form. Furthermore, step 4 calculates d but doesn't conclude the directrix equation. The stated answer is correct, but the solution text is sloppy/misleading regarding the trig function in the standard form description.gpt-oss:20b: fail (error) 2026-10-16 — The first sentence incorrectly states the standard form as r = ed/(1 ± e cosθ) and claims to divide by the constant in the denominator. The given polar equation involves sinθ, so the correct standard form is r = ed/(1 ∓ e sinθ) after dividing by 3. This misstates the model and would mislead a student.qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrect. The standard form is r = ed / (1 - e sin theta), which corresponds to a directrix at y = -d. Since d = 9/8, the directrigpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence incorrectly states the standard form uses "cos θ" instead of "sin θ", which could mislead a student about the correct polar equation for a conic with a horizontal directrix.
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.