Conics in polar form
Problem 8.317 · medium
For the conic \( \displaystyle r = \frac{2}{1 - 5 \sin{\left(\theta \right)}} \), 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).
- \[ \frac{2}{1 \left(- \frac{5 \sin{\left(\theta \right)}}{1} + 1\right)} = \frac{2}{1 - 5 \sin{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-5}\right|}{1} = 5 \]e = 5: > 1, a hyperbola.✓ Proved
- \[ \frac{2}{1 \cdot 5} = \frac{2}{5} \]ed = 2, so d = 2/5.✓ Proved
Answer \( e = 5,\ \text{hyperbola},\ \text{directrix } y = -2/5 \)
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 (misleading) — The first sentence incorrectly states the standard polar form (using cos θ and a ± sign) and claims a division step that is unnecessary. It misleads a student about how to identify the conic’s parameters from the given equation.qwen3.6:27b-mlx: fail (error) — The directrix is identified as y = -2/5, but for the form r = ed / (1 - e sin theta), the directrix is y = -d. Since d = 2/5, the directrix should be y = -2/5. Wait, let me re-verify. Standard form r = ed / (1 - e sin theta) corresponds to directrix y = -d. Here e=5, ed=2 => d=2/5. So directrix is y = -2/5. The stated answer says y = -2/5. The solution calculates d=2/5 but does not explicitly state the sign convention for the directrix relative to the sine term. 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 setup error in the sentence description, although the subsequent algebra handles the sine correctly. More critically, does the solution explicitly link the minus sign in front of sine to the negative y-directrix? Step 4 just calculates d. It fails to justify the sign of the directrix equation. A student might think d is just the distance and forget the sign depends on the trig function and its sign. The sentence in step 1 is also factually incorrect regarding the form (mentions cos instead of sin). This is misleading/error.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The directrix is identified as y = -2/5, but for the form r = ed / (1 - e sin theta), the directrix is y = -d. Since d = 2/5, the directrix should be y = -2/5. Wait, let me re-verify. Standard form r = ed / (1 - e sin theta) corresponds to directrix y = -d. Here e=5, ed=2 => d=2/5. So directrix is y = -2/5. The stated answer says y = -2/5. The solution calculates d=2/5 but does not explicitly state the sign convention for the directrix relative to the sine term. 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 setup error in the sentence description, although the subsequent algebra handles the sine correctly. More critically, does the solution explicitly link the minus sign in front of sine to the negative y-directrix? Step 4 just calculates d. It fails to justify the sign of the directrix equation. A student might think d is just the distance and forget the sign depends on the trig function and its sign. The sentence in step 1 is also factually incorrect regarding the form (mentions cos instead of sin). This is misleading/error.gpt-oss:20b: fail (misleading) 2026-10-17 — The first sentence incorrectly states the standard polar form (using cos θ and a ± sign) and claims a division step that is unnecessary. It misleads a student about how to identify the conic’s parameters from the given equation.qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is identified as y = -2/5, but for the form r = ed/(1 - e sin(theta)), the directrix is y = -d. Since d = 2/5, the directrix should be y = -2/5. Wait, let me re-evaluate. Standard form r = ed / (1 - e sin theta) corresponds to directrix y = -d. Here ed = 2 and e = 5, so d = 2/5. Thus directrix is y = -2/5. The stated answer says y = -2/5. The solution steps calculate d=2/5 but do not explicitly state the sign convention for the directrix relative to the denominator term. However, looking closely at step 4, it just calculates d. It does not state the final directrix equation. The 'Stated answer' is separate from the solution steps. The solution steps end at calculating d. Does the solution conclude the directrix? No. But usually, these reviews check if the steps lead to the answer. The steps are algebraically correct for finding e and d. The identification of the conic is correct. The only potential issue is if the sign of the directrix is mishandled in the student's mind, but the steps shown are just finding d. Let's look at the standard forms again. r = ep / (1 - e sin theta) -> directrix y = -p. Here p = d = 2/5. So y = -2/5. The stated answer is correct. The steps are correct. Why would it fail? Let's re-read carefully. Step 1 says 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. This is a minor error in the description of the standard form in sentence 1, but the subsequent steps use sin. Step 3 calculates e = 5. Step 4 calculates d = 2/5. The solution does not explicitly write the final directrix equation, but derives the components. Is there a deeper error? For r = 2 / (1 - 5 sin theta), e = 5. Since e > 1, it is a hyperbola. The term is -5 sin theta, which matches 1 - e sin theta. The directrix for 1 - e sin theta is y = -d. d = 2/5. So y = -2/5. The stated answer is correct. The steps are correct. Sentence 1 mentions 'cos theta' but the problem is 'sin theta'. This is a 'style' or 'misleading' issue if it implies the form must be cosine. But it says '1 +/- e cos theta' as a generic template? No, it's specific. It should say sin. However, usually 'error' is reserved for mathematical mistakes. Let's check if the directrix sign is actually y = +2/5? Standard polar forms: r = ed / (1 + e sin theta) -> y = d r = ed / (1 - e sin theta) -> y = -d r = ed / (1 + e cos theta) -> x = d r = ed / (1 - e cos theta) -> x = -d Here we have 1 - 5 sin theta. So y = -d = -2/5. The stated answer is y = -2/5. The solution is correct. Why might it be a fail? Maybe the prompt implies the solution *is* the text provided. Sentence 1: 'Divide top and bottom ... to get r = ed/(1 +/- e cos theta)'. This is factually incorrect regarding the variable (sin vs cos). It sets up the wrong template. Even though the subsequent steps fix it, the sentence is wrong. Severity: misleading or style? It's a wrong statement about the standard form. Let's call it 'misleading' because a student might think the standard form only involves cosine. Or is it an 'error'? The sentence claims the result of the division is a cosine form. It is not. I will mark it as 'misleading' because the algebraic steps are correct, but the explanatory sentence is flawed. Actually, looking at similar examples, if the sentence describes the method incorrectly, it's often 'misleading'. Let's double check if there's a calculation error. 2 / (1 - 5 sin theta). e = 5. ed = 2 => d = 2/5. Directrix y = -2/5. Everything is correct except Sentence 1 says 'cos'. I will output fail, misleading.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly claims to divide by a constant in the denominator and rewrites the equation in the form r = ed/(1 ± e cos θ), whereas the given polar equation involves sin θ, not cos θ, and no non‑unit constant needs to be divided out. This misstates the standard form and would mislead a student about how to identify the conic from the polar equation.qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The solution fails to determine the correct sign and orientation of the directrix. For the form r = ed / (1 - e sin(theta)), the directrix is y = -d. Since d = 2/5, the directrix is y = -2/5. However, the provided answer states y = -2/5, which is actually correct for this specific form? Wait. Let's re-verify standard forms. Standard forms: 1. r = ed / (1 + e cos theta) -> x = d 2. r = ed / (1 - e cos theta) -> x = -d 3. r = ed / (1 + e sin theta) -> y = d 4. r = ed / (1 - e sin theta) -> y = -d The given equation is r = 2 / (1 - 5 sin theta). This matches form 4 with e=5 and ed=2 => d=2/5. So the directrix is y = -d = -2/5. The stated answer is y = -2/5. The solution steps calculate d=2/5 but do not explicitly state the rule for the sign/orientation, merely calculating d. Step 4 says 'ed=2, so d=2/5'. It does not conclude the directrix equation. The 'Stated answer' is correct, but the solution text provided in steps 1-4 does not actually derive the directrix equation y = -2/5, it only finds d. More importantly, usually, these problems require identifying the directrix equation. The solution stops at finding d. Is that an error? The prompt asks to 'find the directrix'. Finding 'd' is not finding the directrix. The directrix is a line. The solution fails to state the directrix equation. Actually, looking closer at the provided 'Stated answer' vs the 'Solution'. The solution steps 1-4 are just calculations. Step 4 ends with 'd=2/5'. It does not state the directrix. The user provided 'Stated answer' separately. The review is of the 'Solution' lines. The solution lines do not produce the final answer for the directrix. They only produce d. This is an incomplete solution. However, often in these automated checks, if the intermediate steps are correct and lead to the answer, it might pass. But here, the crucial link between 'd' and the line equation 'y = -d' is missing in the text. A student reading this would not know if the directrix is y=2/5, y=-2/5, x=2/5, etc. The sign depends on the denominator term. The solution does not explain this. This is misleading/incomplete. Let's look at the severity. Is it an error? The calculation of d is correct. The identification of e and conic is correct. The missing step is the final translation of d to the directrix equation. If the 'Stated answer' is considered part of the solution output, then the solution is just missing the explanatory sentence. If the 'Solution' block is all that is reviewed, it is incomplete. Let's assume the 'Stated answer' is the final output and the 'Solution' is the work. The work is incomplete because it doesn't derive the directrix equation. But is it 'wrong'? No. Is it 'misleading'? Yes, because it doesn't show how the sign is determined. Wait, let's look at the standard form again. r = ep / (1 - e sin theta). Directrix is y = -p. Here e=5, ep=2 -> p=2/5. Directrix y = -2/5. The solution calculates p (called d) = 2/5. It does not state y = -2/5. If I mark it 'fail', it's because the solution doesn't answer the question 'find the directrix'. It only finds a parameter. Let's check if there is a more subtle error. Step 1: 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. The problem has sin, not cos. The sentence says 'cos theta' as a generic placeholder? Or is it a typo? It says '1 +/- e cos theta'. The problem is sin. This is a slight inaccuracy in the description of the standard form, but the subsequent steps use sin. The main issue is that the solution does not explicitly state the directrix equation. However, usually, these 'review' tasks look for mathematical errors in the steps provided. The steps provided are algebraically correct for finding e and d. The failure to write 'y = -2/5' might be considered a 'style' or 'incomplete' issue, but often 'error' is reserved for wrong math. Let's look at the 'Stated answer' again. It is correct. If the solution is just the 4 lines, it is incomplete. However, often in these datasets, the 'Solution' is expected to lead to the answer. Let's consider if the sign is wrong. If the form was 1 + e sin, directrix is y = d. If the form was 1 - e sin, directrix is y = -d. The solution does not mention this rule. I will mark it as 'fail' with 'misleading' or 'error' because it doesn't actually find the directrix, only the distance d. A student might think d is the directrix (which is a number, not a line) or might not know how to convert d to the line equation. Actually, looking at similar examples, if the final answer is provided separately, the solution steps are judged on their correctness. The steps are correct. The omission of the final sentence might be 'style'. But 'find the directrix' implies stating the equation. Let's look at Step 1 again. 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. This is a generic statement. It's fine. I'll go with 'fail' 'misleading' because the solution does not explain how the directrix equation is derived from d, specifically the sign and orientation (y vs x, positive vs negative), which is the core of the 'find the directrix' part of the problem. Without this, the student learns nothing about the directrix's position relative to the pole. Alternatively, is there an error in the calculation? e = 5. Correct. d = 2/5. Correct. Directrix y = -2/5. Correct. The solution is just incomplete. In many contexts, incomplete is 'fail'. Let's try to be precise. The prompt asks to find the directrix. The solution finds d. It does not find the directrix. Therefore, the solution does not solve the problem. Verdict: fail. Severity: error (it fails to perform the required task). Wait, 'error' usually means a step is wrong. 'Misleading' means it teaches something false. 'Style' means it's correct but unclear. If I say 'error', I imply the math is wrong. The math isn't wrong, it's just stopped early. If I say 'misleading', I imply it teaches a wrong concept. It doesn't teach a wrong concept, it just omits the final step. Let's look at the provided 'Stated answer'. It is correct. Maybe the 'Solution' is just the derivation of parameters? Let's assume the question implies the solution should contain the final answer. I will mark it as 'fail' 'error' because the solution does not yield the directrix equation, which is part of the question. Actually, let's look at the first sentence again. 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. The problem has sin. The sentence says cos. This is a mismatch. It's a minor typo in the sentence, but it's technically incorrect to say you get a cos form from a sin problem. I'll stick with the main issue: the solution does not state the directrix equation. Verdict: fail. Severity: error. Notes: The solution calculates the parameter d but fails to state the equation of the directrix (y = -2/5), thus not fully answering the problem. Additionally, Step 1 incorrectly references cosine instead of sine in the standard form description.gpt-oss:20b: pass 2026-10-14
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.