Conics in polar form
Problem 8.436 · medium
For the conic \( \displaystyle r = \frac{3}{5 - 8 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{3}{5 \left(1 - \frac{8 \cos{\left(\theta \right)}}{5}\right)} = \frac{3}{5 - 8 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-8}\right|}{5} = \frac{8}{5} \]e = 8/5: > 1, a hyperbola.✓ Proved
- \[ \frac{3}{5 \cdot \frac{8}{5}} = \frac{3}{8} \]ed = 3/5, so d = 3/8.✓ Proved
Answer \( e = \frac{8}{5},\ \text{hyperbola},\ \text{directrix } x = -3/8 \)
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: fail (error) — The directrix sign is wrong: for r = ed/(1‑e cosθ) the directrix is x = +d, not x = –d. The solution incorrectly states x = –3/8.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 = 3/8, so the directrix is x = -3/8. 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 solution has 1 - (8/5)cos(theta), so it matches 1 - e cos theta with e=8/5. Thus the directrix is x = -d = -3/8. The stated answer says x = -3/8. The solution calculates d=3/8 but does not explicitly state the sign rule for the directrix, implying the student might just take d. However, the final stated answer in the problem description is x = -3/8. Let's look at the solution steps again. Step 4 calculates d=3/8. It does not conclude the directrix equation. The problem asks to find the directrix. The solution stops at finding d. It fails to state the final directrix equation. Also, the prompt says 'Stated answer: ... directrix x = -3/8'. The solution provided in the text ends at step 4. It does not provide the final answer for the directrix. This is an incomplete solution. Furthermore, usually, one must explicitly link the form to the directrix location. The solution is incomplete.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The directrix is incorrect. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d. Here d = 3/8, so the directrix is x = -3/8. 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 solution has 1 - (8/5)cos(theta), so it matches 1 - e cos theta with e=8/5. Thus the directrix is x = -d = -3/8. The stated answer says x = -3/8. The solution calculates d=3/8 but does not explicitly state the sign rule for the directrix, implying the student might just take d. However, the final stated answer in the problem description is x = -3/8. Let's look at the solution steps again. Step 4 calculates d=3/8. It does not conclude the directrix equation. The problem asks to find the directrix. The solution stops at finding d. It fails to state the final directrix equation. Also, the prompt says 'Stated answer: ... directrix x = -3/8'. The solution provided in the text ends at step 4. It does not provide the final answer for the directrix. This is an incomplete solution. Furthermore, usually, one must explicitly link the form to the directrix location. The solution is incomplete.gpt-oss:20b: fail (error) 2026-10-18 — The directrix sign is wrong: for r = ed/(1‑e cosθ) the directrix is x = +d, not x = –d. The solution incorrectly states x = –3/8.qwen3.6:27b-mlx: fail (error) 2026-10-16 — The directrix is incorrectly identified as x = -3/8. Since the denominator contains (1 - e cos theta) (after factoring out the negative sign or observing the -8 cos theta term), the directrix is vertical and located at x = -d, which is x = -3/8? Wait. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here we have r = (3/5) / (1 - (8/5)cos theta). So ed = 3/5, e = 8/5, d = 3/8. The form is 1 - e cos theta, which implies the directrix is x = -d = -3/8. Actually, let's re-verify. Standard forms: 1. r = ed / (1 + e cos theta) -> Directrix x = d 2. r = ed / (1 - e cos theta) -> Directrix x = -d 3. r = ed / (1 + e sin theta) -> Directrix y = d 4. r = ed / (1 - e sin theta) -> Directrix y = -d The given equation is r = 3 / (5 - 8 cos theta). Divide by 5: r = (3/5) / (1 - (8/5) cos theta). This matches form 2: 1 - e cos theta with e = 8/5. So the directrix is x = -d. d = (3/5) / (8/5) = 3/8. So directrix is x = -3/8. The stated answer says directrix x = -3/8. The solution calculates d = 3/8 but does not explicitly state the sign logic for the directrix location, merely calculating d. However, the final answer provided in the problem statement is x = -3/8. The solution steps end at calculating d=3/8. It does not explicitly conclude the directrix equation. But usually, these reviews check if the solution *leads* to the answer. The solution stops at d=3/8. It doesn't state the directrix. Is that an error? The prompt asks to find the directrix. The solution calculates d but doesn't write the final equation. However, looking at the provided 'Stated answer', it matches. The issue is likely that the solution text doesn't explicitly link d=3/8 to x=-3/8. But is it 'wrong'? Let's look closer. Step 1 says 'Divide ... to get r = ed/(1 ± e cos theta)'. Step 2 shows the algebra. Step 3 finds e. Step 4 finds d. It never states the directrix. This is an incomplete solution. But is it 'wrong'? Actually, there is a subtle sign error potential. r = 3 / (5 - 8 cos theta) = (3/5) / (1 - 1.6 cos theta). This is of the form ed / (1 - e cos theta). For this form, the directrix is x = -d. d = 3/8. So x = -3/8. The solution calculates d=3/8. It does not state the directrix. If the 'Stated answer' is the target, the solution fails to produce it explicitly. However, often in these formats, if the intermediate steps are correct, it might pass. But wait, look at Step 1: 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos theta)'. This is a bit vague on the sign. Let's check the severity. If I say 'fail', what is the error? The solution doesn't actually answer the third part of the question (find the directrix) in the text provided, it just calculates d. A student might think d is the directrix coordinate, missing the sign and axis. This is misleading/incomplete. However, there is a more critical check. Is the directrix x = -3/8 correct? Yes. Does the solution derive it? No, it stops at d. Is this an 'error' or 'misleading'? It's an incomplete solution. Let's look for a harder error. What if the form was interpreted as 1 + e cos(-theta)? No. Let's assume the question implies the solution *is* the text provided. The text provided does not state the directrix. Therefore, it fails to answer the problem. Another possibility: The sign of the directrix. If the student thinks r = ed/(1-e cos theta) means x=d, they would get x=3/8. The correct is x=-3/8. The solution calculates d=3/8. It does not specify the sign. This is a common pitfall. By not stating the rule for the sign, the solution is misleading/incomplete. I will mark it as fail because it does not explicitly state the directrix equation, leaving the sign ambiguous to a student who doesn't know the rule.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The directrix is incorrectly identified as x = -3/8. For the form r = ed/(1 - e cos θ), the directrix is x = -d. Here, the denominator is 1 - (8/5)cos θ, so e = 8/5 and d = 3/8, making the directrix x = -3/8. Wait, let me re-evaluate. The standard form is r = ed / (1 ± e cos θ). The given equation is r = 3 / (5 - 8 cos θ) = (3/5) / (1 - (8/5) cos θ). This matches the form with a minus sign: 1 - e cos θ. For this form, the directrix is x = -d. Since ed = 3/5 and e = 8/5, d = 3/8. Thus, the directrix is x = -3/8. The stated answer says x = -3/8. The solution calculates d = 3/8 but does not explicitly state the sign rule for the directrix location relative to the pole, which is crucial. However, looking closely at step 4, it just calculates d. It doesn't state the final directrix equation in the steps, but the 'Stated answer' has it. The prompt asks to review the *solution*. The solution stops at calculating d. It fails to explicitly connect d to the directrix equation x = -d (or x = d). Without stating the rule, the jump to the final answer (if it were included) would be unjustified. But more importantly, is the sign correct? r = ed/(1 - e cos θ) corresponds to directrix x = -d. Yes. So x = -3/8 is correct. The solution is incomplete because it calculates d but doesn't state the directrix equation. Is that an error? The problem asks to 'find the directrix'. The solution ends at step 4 calculating d. It does not state the directrix. This is an incomplete solution. However, usually, these reviews look for mathematical errors. Let's look at the 'Stated answer' vs 'Solution'. The solution provided is just the steps. Step 4 finds d. It does not conclude with the directrix equation. This is a 'fail' because the solution does not answer the full question (finding the directrix). It only finds d. A student might think d is the directrix. This is misleading/incomplete.gpt-oss:20b: pass 2026-10-15
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.