Conics in polar form
Problem 8.510 · medium
For the conic \( \displaystyle r = \frac{1}{5 \cos{\left(\theta \right)} + 5} \), 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).
- \[ \frac{1}{5 \left(\cos{\left(\theta \right)} + 1\right)} = \frac{1}{5 \cos{\left(\theta \right)} + 5} \]The standard form.✓ Proved
- \[ \frac{\left|{5}\right|}{5} = 1 \]e = 1: = 1, a parabola.✓ Proved
- \[ 1 \cdot \frac{1}{5} \cdot 1^{-1} = \frac{1}{5} \]ed = 1/5, so d = 1/5.✓ Proved
Answer \( e = 1,\ \text{parabola},\ \text{directrix } x = 1/5 \)
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: passqwen3.6:27b-mlx: fail (error) — The solution fails to identify the correct directrix. For the form r = ed / (1 + e cos theta), the directrix is x = d. Here d = 1/5, so the directrix is x = 1/5. However, the standard form derivation in step 2 is algebraically messy and the logic in step 3/4 is presented as isolated equations rather than a coherent derivation. More critically, while the final answer x=1/5 is actually correct for this specific equation (since the + sign implies the directrix is at x = +d), the explanation is extremely opaque. Wait, let me re-evaluate. r = 1/(5cos(theta)+5) = (1/5) / (cos(theta)+1). Standard form is r = ed / (1 + e cos theta). Here e=1, ed = 1/5 => d=1/5. Since it is 1 + e cos theta, the directrix is x = d = 1/5. The answer is correct. Why did I think it was wrong? Let me check the sign convention. r = ep / (1 + e cos theta) corresponds to directrix x = p. Yes. So the answer is correct. Is the solution misleading? Step 2 shows an identity that doesn't really 'get' the standard form, it just verifies the original equation. Step 3 calculates Abs(5)/5 which is 1, claiming e=1. This is a heuristic that works here but isn't a rigorous derivation. Step 4 calculates d. The main issue is that the 'sentences' are just labels for equations that don't clearly explain the mapping to the standard form. Specifically, Step 1 says 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. The equation in Step 2 does NOT show this division clearly resulting in the standard form structure explicitly identifying e and d. It just shows an equality. A student might not learn how to extract e and d. However, is it 'wrong'? The final answer is correct. The steps are 'proved' equations. The sentences are weak. Let's look closer at Step 3: 'Abs(5)/5 = 1 -- e = 1'. This is a non-sequitur unless explained. The coefficient of cos(theta) in the denominator after normalizing the constant term to 1 is e. Here, dividing by 5 gives (1/5)/(cos(theta)+1). The coefficient of cos is 1, so e=1. The solution writes 'Abs(5)/5', which is weird. It seems to be taking the coefficient of cos (5) and dividing by the constant term (5). That is a valid method to find e. But the presentation is very poor. Is it 'misleading'? It might teach a student a weird trick 'Abs(coeff)/constant' without understanding the normalization. But the verdict should be 'fail' if it teaches something false or is an error. It's not an error in the final result. It's 'misleading' or 'style'. Let's look at the directrix again. r = ed/(1+ecos(theta)). Directrix is x=d. Here d=1/5. Correct. What if the form was 1 - ecos(theta)? Then x=-d. The solution doesn't discuss the sign. But the sign is positive. I will mark it as 'misleading' because the derivation of e and d is presented as magic arithmetic steps rather than algebraic manipulation into standard form, which could confuse a student about how to handle other cases.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to identify the correct directrix. For the form r = ed / (1 + e cos theta), the directrix is x = d. Here d = 1/5, so the directrix is x = 1/5. However, the standard form derivation in step 2 is algebraically messy and the logic in step 3/4 is presented as isolated equations rather than a coherent derivation. More critically, while the final answer x=1/5 is actually correct for this specific equation (since the + sign implies the directrix is at x = +d), the explanation is extremely opaque. Wait, let me re-evaluate. r = 1/(5cos(theta)+5) = (1/5) / (cos(theta)+1). Standard form is r = ed / (1 + e cos theta). Here e=1, ed = 1/5 => d=1/5. Since it is 1 + e cos theta, the directrix is x = d = 1/5. The answer is correct. Why did I think it was wrong? Let me check the sign convention. r = ep / (1 + e cos theta) corresponds to directrix x = p. Yes. So the answer is correct. Is the solution misleading? Step 2 shows an identity that doesn't really 'get' the standard form, it just verifies the original equation. Step 3 calculates Abs(5)/5 which is 1, claiming e=1. This is a heuristic that works here but isn't a rigorous derivation. Step 4 calculates d. The main issue is that the 'sentences' are just labels for equations that don't clearly explain the mapping to the standard form. Specifically, Step 1 says 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. The equation in Step 2 does NOT show this division clearly resulting in the standard form structure explicitly identifying e and d. It just shows an equality. A student might not learn how to extract e and d. However, is it 'wrong'? The final answer is correct. The steps are 'proved' equations. The sentences are weak. Let's look closer at Step 3: 'Abs(5)/5 = 1 -- e = 1'. This is a non-sequitur unless explained. The coefficient of cos(theta) in the denominator after normalizing the constant term to 1 is e. Here, dividing by 5 gives (1/5)/(cos(theta)+1). The coefficient of cos is 1, so e=1. The solution writes 'Abs(5)/5', which is weird. It seems to be taking the coefficient of cos (5) and dividing by the constant term (5). That is a valid method to find e. But the presentation is very poor. Is it 'misleading'? It might teach a student a weird trick 'Abs(coeff)/constant' without understanding the normalization. But the verdict should be 'fail' if it teaches something false or is an error. It's not an error in the final result. It's 'misleading' or 'style'. Let's look at the directrix again. r = ed/(1+ecos(theta)). Directrix is x=d. Here d=1/5. Correct. What if the form was 1 - ecos(theta)? Then x=-d. The solution doesn't discuss the sign. But the sign is positive. I will mark it as 'misleading' because the derivation of e and d is presented as magic arithmetic steps rather than algebraic manipulation into standard form, which could confuse a student about how to handle other cases.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to identify the directrix as a line (x = 1/5), instead only calculating the distance d. Furthermore, the standard form derivation is ambiguous regarding the sign of the cosine term, which determines whether the directrix is at x = +d or x = -d.gpt-oss:20b: fail (misleading) 2026-10-18 — The step claiming e = |5|/5 is not justified by the standard form; it gives the correct value but misleads a student about how to extract e from the equation.qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly manipulates the equation into standard polar form, identifies the eccentricity and conic type, and correctly calculates the directrix.gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly derives the eccentricity by taking |5|/5, which is not justified. The correct reasoning is that the denominator is 5(1+cosθ), so e=1 directly from the form r=ed/(1+e cosθ).
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.