Conics in polar form
Problem 8.311 · medium
For the conic \( \displaystyle r = \frac{10}{6 \sin{\left(\theta \right)} + 1} \), 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).
- \[ \frac{10}{1 \left(\frac{6 \sin{\left(\theta \right)}}{1} + 1\right)} = \frac{10}{6 \sin{\left(\theta \right)} + 1} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{1} = 6 \]e = 6: > 1, a hyperbola.✓ Proved
- \[ \frac{10}{1 \cdot 6} = \frac{5}{3} \]ed = 10, so d = 5/3.✓ Proved
Answer \( e = 6,\ \text{hyperbola},\ \text{directrix } y = 5/3 \)
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 (error) — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r=ed/(1±e cosθ); the given equation involves sinθ, not cosθ, and the manipulation is not justified. The subsequent identification of e=6 and d=5/3 ignores the correct sign conventions for the directrix of a hyperbola with a vertical axis, leading to an incorrect directrix.qwen3.6:27b-mlx: fail (error) — The solution incorrectly identifies the directrix as y = 5/3. For a conic in the form r = ed / (1 + e sin theta), the directrix is the horizontal line y = d. Here, d = 5/3, so the directrix is y = 5/3. Wait, let me re-read the standard forms. Standard form: r = ed / (1 + e sin theta) corresponds to a directrix y = d (if the sign is +) or y = -d (if the sign is -). Let's check the derivation. r = 10 / (1 + 6 sin theta). This matches r = ed / (1 + e sin theta) with e=6 and ed=10 => d=5/3. The directrix for r = ed / (1 + e sin theta) is y = d. So y = 5/3. Wait, is the directrix y=d or y=-d? Definition: r = e * distance_to_point / distance_to_line. If directrix is y = d, distance is |d - r sin theta|. r = e |d - r sin theta|. If we assume the focus is at the origin and the directrix is above the focus (y=d > 0), then for points on the conic closer to the focus than the directrix (ellipse) or further (hyperbola), we need to be careful with signs. Standard polar equation derivation: Let directrix be y = d. r = e (d - r sin theta) => r(1 + e sin theta) = ed => r = ed / (1 + e sin theta). So yes, r = ed / (1 + e sin theta) corresponds to directrix y = d. Let's double check the other case. If directrix is y = -d. r = e (r sin theta - (-d)) = e(r sin theta + d) => r(1 - e sin theta) = ed => r = ed / (1 - e sin theta). The given equation is r = 10 / (1 + 6 sin theta). This matches the form with + sin theta. So the directrix is y = d = 5/3. The stated answer says: directrix y = 5/3. The solution steps say: d = 5/3. It does not explicitly state the equation of the directrix in the text, but the final answer provided in the prompt is "directrix y = 5/3". Is there an error? Let's check the eccentricity. e = 6. Correct. Conic: Hyperbola. Correct. Directrix: y = 5/3. Correct. Why would this fail? Let's look at the sentences. Sentence 1: "Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ)." The formula given in the sentence uses cos theta, but the problem has sin theta. This is a mismatch in the general formula stated. It should say "1 ± e sin θ" or "1 ± e cos θ" depending on the term. Stating only cos is misleading/wrong for this specific problem which uses sin. Also, the directrix location depends on the sign. r = ed / (1 + e sin theta) -> Directrix y = d. r = ed / (1 - e sin theta) -> Directrix y = -d. The solution calculates d = 5/3. The final answer claims y = 5/3. This is correct for the + sign. However, Sentence 1 says "get r = ed/(1 ± e cos θ)". This is factually incorrect as a general statement for *this* problem which involves sine. It implies the standard form only involves cosine, or it's a typo. If a student reads this, they might think they must convert sin to cos or that the formula doesn't apply to sine. Let's look closer. Sentence 1: "Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ)." This sentence is wrong because the problem has sin(theta), not cos(theta). The standard form for sine is r = ed/(1 ± e sin θ). By stating the target form has cos, the sentence is mathematically inaccurate for the context. Therefore, the verdict is fail, severity misleading/error. The sentence describes the wrong standard form (cosine instead of sine).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the directrix as y = 5/3. For a conic in the form r = ed / (1 + e sin theta), the directrix is the horizontal line y = d. Here, d = 5/3, so the directrix is y = 5/3. Wait, let me re-read the standard forms. Standard form: r = ed / (1 + e sin theta) corresponds to a directrix y = d (if the sign is +) or y = -d (if the sign is -). Let's check the derivation. r = 10 / (1 + 6 sin theta). This matches r = ed / (1 + e sin theta) with e=6 and ed=10 => d=5/3. The directrix for r = ed / (1 + e sin theta) is y = d. So y = 5/3. Wait, is the directrix y=d or y=-d? Definition: r = e * distance_to_point / distance_to_line. If directrix is y = d, distance is |d - r sin theta|. r = e |d - r sin theta|. If we assume the focus is at the origin and the directrix is above the focus (y=d > 0), then for points on the conic closer to the focus than the directrix (ellipse) or further (hyperbola), we need to be careful with signs. Standard polar equation derivation: Let directrix be y = d. r = e (d - r sin theta) => r(1 + e sin theta) = ed => r = ed / (1 + e sin theta). So yes, r = ed / (1 + e sin theta) corresponds to directrix y = d. Let's double check the other case. If directrix is y = -d. r = e (r sin theta - (-d)) = e(r sin theta + d) => r(1 - e sin theta) = ed => r = ed / (1 - e sin theta). The given equation is r = 10 / (1 + 6 sin theta). This matches the form with + sin theta. So the directrix is y = d = 5/3. The stated answer says: directrix y = 5/3. The solution steps say: d = 5/3. It does not explicitly state the equation of the directrix in the text, but the final answer provided in the prompt is "directrix y = 5/3". Is there an error? Let's check the eccentricity. e = 6. Correct. Conic: Hyperbola. Correct. Directrix: y = 5/3. Correct. Why would this fail? Let's look at the sentences. Sentence 1: "Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ)." The formula given in the sentence uses cos theta, but the problem has sin theta. This is a mismatch in the general formula stated. It should say "1 ± e sin θ" or "1 ± e cos θ" depending on the term. Stating only cos is misleading/wrong for this specific problem which uses sin. Also, the directrix location depends on the sign. r = ed / (1 + e sin theta) -> Directrix y = d. r = ed / (1 - e sin theta) -> Directrix y = -d. The solution calculates d = 5/3. The final answer claims y = 5/3. This is correct for the + sign. However, Sentence 1 says "get r = ed/(1 ± e cos θ)". This is factually incorrect as a general statement for *this* problem which involves sine. It implies the standard form only involves cosine, or it's a typo. If a student reads this, they might think they must convert sin to cos or that the formula doesn't apply to sine. Let's look closer. Sentence 1: "Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ)." This sentence is wrong because the problem has sin(theta), not cos(theta). The standard form for sine is r = ed/(1 ± e sin θ). By stating the target form has cos, the sentence is mathematically inaccurate for the context. Therefore, the verdict is fail, severity misleading/error. The sentence describes the wrong standard form (cosine instead of sine).gpt-oss:20b: fail (error) 2026-10-17 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r=ed/(1±e cosθ); the given equation involves sinθ, not cosθ, and the manipulation is not justified. The subsequent identification of e=6 and d=5/3 ignores the correct sign conventions for the directrix of a hyperbola with a vertical axis, leading to an incorrect directrix.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to explicitly state the equation of the directrix (y = 5/3), only calculating the distance d. Additionally, the setup sentence is misleading as it references 'cos' while the problem involves 'sin', and does not explain how the sign in the denominator determines the position of the directrix.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly claims that dividing by the constant in the denominator yields the standard form r = ed/(1 ± e cosθ). The given polar equation contains sinθ, so the correct standard form should involve e sinθ, not e cosθ. This misstatement misleads the reader about the relationship between the given equation and the standard conic form.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to divide the denominator by the constant term (1) to normalize the constant to 1, leaving the equation in a non-standard form. Consequently, it incorrectly identifies the coefficient of sin(theta) (6) as the eccentricity, whereas the correct eccentricity is 6/1 = 6 only if the constant is 1, but the standard form requires the constant term in the denominator to be 1. Wait, the constant IS 1. Let me re-read. r = 10 / (6 sin theta + 1). Standard form is r = ed / (1 + e sin theta). Here, the denominator is 1 + 6 sin theta. So e = 6. And ed = 10 => 6d = 10 => d = 5/3. The eccentricity is indeed 6. The conic is a hyperbola. The directrix is y = 5/3. The solution steps are actually correct algebraically. Step 1 says 'Divide top and bottom by the constant...'. The constant is 1. Dividing by 1 changes nothing. Step 2 shows the equation. Step 3 calculates e = 6/1 = 6. Step 4 calculates d = 10/6 = 5/3. The logic holds. Why did I think it was wrong? Ah, I misread the prompt's implied error. Let's look closer. Is there a subtle error? 'Divide top and bottom by the constant in the denominator'. The constant is 1. The resulting form is r = 10 / (1 + 6 sin theta). This matches r = ed / (1 + e sin theta). So e=6, ed=10. This is correct. The verdict should be pass? No, wait. The prompt asks me to find what is wrong. Is there a convention issue? Usually, standard form is written with the constant 1 first. The given equation has 1 last. The solution doesn't explicitly rewrite it as 1 + 6 sin theta, but Step 3 implies it by taking Abs(6)/1. This is a bit loose but mathematically sound. However, looking at the provided solution text again: Step 1 says 'Divide... to get r = ed/(1 ± e cos theta)'. It mentions cos theta, but the problem has sin theta. This is a mismatch in the stated method vs the problem type. The sentence claims the target form involves cos, but the problem involves sin. This is a 'misleading' or 'style' error in the sentence description, but does it affect the result? The result is correct. But the sentence is factually incorrect about the form it is aiming for (cos vs sin). A student might be confused why they are looking for cos when they have sin. I will mark it as misleading because the sentence states the target form incorrectly for this specific problem.gpt-oss:20b: fail (error) 2026-10-14 — The first sentence incorrectly states the standard polar form for a conic with a horizontal directrix. It claims r = ed/(1 ± e cosθ) after dividing by the constant, but the given equation involves sinθ, so the correct form is r = ed/(1 ± e sinθ). This misstatement could mislead a student about the relationship between the denominator and the directrix orientation.
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.