Conics in polar form
Problem 8.458 · medium
For the conic \( \displaystyle r = \frac{2}{3 - 7 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{2}{3 \left(1 - \frac{7 \cos{\left(\theta \right)}}{3}\right)} = \frac{2}{3 - 7 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-7}\right|}{3} = \frac{7}{3} \]e = 7/3: > 1, a hyperbola.✓ Proved
- \[ \frac{2}{3 \cdot \frac{7}{3}} = \frac{2}{7} \]ed = 2/3, so d = 2/7.✓ Proved
Answer \( e = \frac{7}{3},\ \text{hyperbola},\ \text{directrix } x = -2/7 \)
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 correctly finds e=7/3 and identifies a hyperbola, but the directrix is incorrectly computed. For r=ed/(1+e cosθ) with e>1, the directrix is x=−d/e, not x=−d. Using d=2/7 gives x=−2/7, which is inconsistent with the derived value of d/e. The conclusion about the directrix is therefore incorrect.qwen3.6:27b-mlx: fail (error) — The directrix is incorrect. For the form r = ed / (1 - e cos theta), the directrix is x = -d. Here d = 2/7, so the directrix is x = -2/7. Wait, let me re-evaluate. The standard form is r = ed / (1 + e cos theta) for x = d, or r = ed / (1 - e cos theta) for x = -d. The given equation is r = 2 / (3 - 7 cos theta) = (2/3) / (1 - (7/3) cos theta). This matches 1 - e cos theta with e = 7/3. Thus the directrix is x = -d = -2/7. The stated answer says x = -2/7. The solution calculates d = 2/7 but does not explicitly state the sign rule for the directrix, potentially leading to confusion, but the final answer in the problem statement is actually correct. However, the solution text itself ends at calculating d. It does not explicitly conclude the directrix equation. The prompt asks to review the solution lines. Line 4 calculates d. It does not state the directrix. The 'Stated answer' is separate. The solution is incomplete because it doesn't state the directrix. But usually, these reviews check for errors in the provided lines. Let's look closer. Line 1 says 'Divide ... to get r = ed/(1 +/- e cos theta)'. Line 2 shows the algebra. Line 3 finds e. Line 4 finds d. The solution fails to explicitly identify the directrix equation, which is part of the problem. Is this an error or just incomplete? The prompt says 'find the directrix'. The solution stops at finding d. This is an incomplete solution. However, often 'error' is reserved for mathematical mistakes. Let's look at the sign. r = ed / (1 - e cos theta). Directrix is x = -d. d = 2/7. So x = -2/7. The stated answer is correct. The solution lines are mathematically correct but incomplete. If I must choose between pass/fail, an incomplete solution that doesn't answer the full prompt is a fail. But is it an 'error' in the lines? No. Is it 'misleading'? No. Is it 'style'? No. Let's re-read the instructions. 'Check that... each conclusion follows... nothing would teach a student something false.' The solution doesn't teach anything false, it just stops short. However, typically in these adversarial reviews, if the final answer isn't derived, it's a fail. Let's look for a subtle error. Line 1: 'Divide top and bottom by the constant...'. The constant is 3. 2/3 and -7/3. Correct. Line 2: Standard form. Correct. Line 3: e = 7/3. Correct. Line 4: d = 2/7. Correct. The missing step is linking d to the directrix equation. I will mark it as fail due to incompleteness, but what severity? 'Error' is usually for wrong math. 'Misleading' for wrong concepts. 'Style' for clarity. Incompleteness is often treated as an error in the context of 'solving the problem'. Let's assume the question implies the solution *is* the text provided. The text provided does not find the directrix. Therefore it fails to solve the problem. I will label it error because it fails to complete the task.
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 cos theta), the directrix is x = -d. Here d = 2/7, so the directrix is x = -2/7. Wait, let me re-evaluate. The standard form is r = ed / (1 + e cos theta) for x = d, or r = ed / (1 - e cos theta) for x = -d. The given equation is r = 2 / (3 - 7 cos theta) = (2/3) / (1 - (7/3) cos theta). This matches 1 - e cos theta with e = 7/3. Thus the directrix is x = -d = -2/7. The stated answer says x = -2/7. The solution calculates d = 2/7 but does not explicitly state the sign rule for the directrix, potentially leading to confusion, but the final answer in the problem statement is actually correct. However, the solution text itself ends at calculating d. It does not explicitly conclude the directrix equation. The prompt asks to review the solution lines. Line 4 calculates d. It does not state the directrix. The 'Stated answer' is separate. The solution is incomplete because it doesn't state the directrix. But usually, these reviews check for errors in the provided lines. Let's look closer. Line 1 says 'Divide ... to get r = ed/(1 +/- e cos theta)'. Line 2 shows the algebra. Line 3 finds e. Line 4 finds d. The solution fails to explicitly identify the directrix equation, which is part of the problem. Is this an error or just incomplete? The prompt says 'find the directrix'. The solution stops at finding d. This is an incomplete solution. However, often 'error' is reserved for mathematical mistakes. Let's look at the sign. r = ed / (1 - e cos theta). Directrix is x = -d. d = 2/7. So x = -2/7. The stated answer is correct. The solution lines are mathematically correct but incomplete. If I must choose between pass/fail, an incomplete solution that doesn't answer the full prompt is a fail. But is it an 'error' in the lines? No. Is it 'misleading'? No. Is it 'style'? No. Let's re-read the instructions. 'Check that... each conclusion follows... nothing would teach a student something false.' The solution doesn't teach anything false, it just stops short. However, typically in these adversarial reviews, if the final answer isn't derived, it's a fail. Let's look for a subtle error. Line 1: 'Divide top and bottom by the constant...'. The constant is 3. 2/3 and -7/3. Correct. Line 2: Standard form. Correct. Line 3: e = 7/3. Correct. Line 4: d = 2/7. Correct. The missing step is linking d to the directrix equation. I will mark it as fail due to incompleteness, but what severity? 'Error' is usually for wrong math. 'Misleading' for wrong concepts. 'Style' for clarity. Incompleteness is often treated as an error in the context of 'solving the problem'. Let's assume the question implies the solution *is* the text provided. The text provided does not find the directrix. Therefore it fails to solve the problem. I will label it error because it fails to complete the task.gpt-oss:20b: fail (error) 2026-10-19 — The solution correctly finds e=7/3 and identifies a hyperbola, but the directrix is incorrectly computed. For r=ed/(1+e cosθ) with e>1, the directrix is x=−d/e, not x=−d. Using d=2/7 gives x=−2/7, which is inconsistent with the derived value of d/e. The conclusion about the directrix is therefore incorrect.qwen3.6:27b-mlx: fail (style) 2026-10-16 — [domain objection, downgraded to style] The directrix is incorrectly identified as x = -2/7. Since the standard form has a negative cosine term (1 - e cos θ), the directrix is to the left of the pole, meaning x = -d, so x = -2/7 is actually correct? Wait. Let's re-evaluate. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here we have r = (2/3) / (1 - (7/3)cos theta). So ed = 2/3, e = 7/3, d = 2/7. The directrix is x = -d = -2/7. The stated answer says x = -2/7. The solution calculates d = 2/7 but does not explicitly state the sign of the directrix equation, just 'd = 2/7'. However, the final stated answer in the problem description is x = -2/7. The solution step 4 concludes 'd = 2/7'. It does not explicitly write the equation of the directrix. But usually, finding the directrix means finding the line equation. If the solution stops at d=2/7, it hasn't fully answered 'find the directrix'. More importantly, let's look at the sign. The denominator is 3 - 7cos(theta). Dividing by 3 gives 1 - (7/3)cos(theta). This is 1 - e cos(theta). This form corresponds to a directrix at x = -d. So the directrix is x = -2/7. The stated answer is correct. The solution calculates d=2/7. It fails to explicitly state the directrix equation x = -2/7, only providing the distance d. This is incomplete. Is it an error? It's a missing step. Let's look closer. Step 4 says 'ed = 2/3, so d = 2/7'. It does not conclude with the directrix equation. The problem asks to 'find the directrix'. Providing just 'd' is ambiguous without the sign/location. A student might think the directrix is x = 2/7. This is misleading/error.gpt-oss:20b: fail (misleading) 2026-10-16 — The solution incorrectly states the directrix as x = –2/7. For r = ed/(1 – e cosθ) with e>0, the directrix is x = +d = 2/7, not –2/7. The sign of the directrix is misapplied, leading to a false conclusion.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is incorrectly identified as x = -2/7. Since the standard form has a negative cosine term (1 - e cos θ), the directrix is vertical and located at x = -d, which is x = -2/7. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos θ) corresponds to directrix x = -d. Here d = 2/7, so x = -2/7. The stated answer says x = -2/7. The solution calculates d=2/7 but does not explicitly state the sign convention for the directrix location relative to the cosine term. However, looking closely at step 1, it says 'Divide top and bottom... to get r = ed/(1 ± e cos θ)'. Step 2 shows the result with +1 in denominator, implying the form 1 - e cos(theta) was used (since -7/3 cos + 1 = 1 - 7/3 cos). For r = ed/(1 - e cos θ), the directrix is x = -d. The solution finds d=2/7. It does not explicitly conclude x = -2/7, but the stated answer has it. The prompt asks to review the *solution* lines. The solution lines stop at finding d. The 'Stated answer' is separate. The solution itself is incomplete as it doesn't state the directrix equation, only d. But more importantly, is the sign correct? If the form is 1 - e cos θ, directrix is x = -d. If the form is 1 + e cos θ, directrix is x = d. The equation is 3 - 7 cos θ. Dividing by 3 gives 1 - (7/3) cos θ. This matches 1 - e cos θ. So directrix is x = -d = -2/7. The stated answer is correct. The solution calculates d correctly. Does it fail? The solution doesn't explicitly write the directrix equation. It just finds d. The problem asks to 'find the directrix'. The solution stops at d. This is an incomplete solution. Also, step 1 is a sentence. Step 2 is an equation. Step 3 is an equation. Step 4 is an equation. The solution fails to explicitly state the directrix equation x = -2/7, only calculating d. This is a 'fail' because it doesn't fully answer the prompt's request to 'find the directrix' in the solution steps provided, or at least it's misleadingly incomplete. However, usually 'find the directrix' implies finding its equation. The solution finds d. Is that enough? No. It needs to specify the line. Also, the sign of the directrix depends on the sign in the denominator. The solution doesn't explain the sign convention. A student might think d is always positive and x=d. Here x=-d. The solution is misleading/incomplete regarding the final answer format.gpt-oss:20b: fail (error) 2026-10-15 — The solution incorrectly assigns the sign of the directrix. For the standard polar form r = ed/(1 – e cosθ) the directrix is x = +d, not –d. Thus the directrix should be x = 2/7, not x = –2/7.
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.