Conics in polar form
Problem 8.440 · medium
For the conic \( \displaystyle r = \frac{4}{3 \sin{\left(\theta \right)} + 4} \), 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).
- \[ \frac{1}{\frac{3 \sin{\left(\theta \right)}}{4} + 1} = \frac{4}{3 \sin{\left(\theta \right)} + 4} \]The standard form.✓ Proved
- \[ \frac{\left|{3}\right|}{4} = \frac{3}{4} \]e = 3/4: < 1, an ellipse.✓ Proved
- \[ \frac{4}{4 \cdot \frac{3}{4}} = \frac{4}{3} \]ed = 1, so d = 4/3.✓ Proved
Answer \( e = \frac{3}{4},\ \text{ellipse},\ \text{directrix } y = 4/3 \)
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 claims the standard form involves ±e cosθ, but the given equation contains sinθ. It also misstates the division step, leading a student to think the form is r=ed/(1±e cosθ) when it is actually r=ed/(1±e sinθ).qwen3.6:27b-mlx: fail (error) — The solution correctly identifies the eccentricity and type of conic, but fails to determine the correct orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = k), not vertical. The calculation d = 4/3 is correct for the distance, but the final answer must specify y = 4/3 (or y = -4/3 depending on sign convention, here positive). The stated answer in the problem description is actually correct (y=4/3), but the solution steps do not explicitly justify why it is y and not x, nor do they check the sign. More critically, step 1 claims to get the form `r = ed/(1 ± e cos θ)`, but the result in step 2 has `sin`. This is a contradiction in the setup sentence. The standard form for sine is `r = ed/(1 ± e sin θ)`. Using the cosine form as the target is an error in the method description.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution correctly identifies the eccentricity and type of conic, but fails to determine the correct orientation of the directrix. Since the denominator contains sin(theta), the directrix is horizontal (y = k), not vertical. The calculation d = 4/3 is correct for the distance, but the final answer must specify y = 4/3 (or y = -4/3 depending on sign convention, here positive). The stated answer in the problem description is actually correct (y=4/3), but the solution steps do not explicitly justify why it is y and not x, nor do they check the sign. More critically, step 1 claims to get the form `r = ed/(1 ± e cos θ)`, but the result in step 2 has `sin`. This is a contradiction in the setup sentence. The standard form for sine is `r = ed/(1 ± e sin θ)`. Using the cosine form as the target is an error in the method description.gpt-oss:20b: fail (misleading) 2026-10-18 — The first sentence incorrectly claims the standard form involves ±e cosθ, but the given equation contains sinθ. It also misstates the division step, leading a student to think the form is r=ed/(1±e cosθ) when it is actually r=ed/(1±e sinθ).qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to identify the correct orientation of the directrix. Since the equation involves sin(theta), the directrix is horizontal (y = d), but the standard form comparison in step 1 implies a generic template without specifying that sin corresponds to horizontal directrices, leading to ambiguity or error if the student assumes cos corresponds to x. More critically, step 4 calculates d=4/3 but does not explicitly state the directrix is y = 4/3, relying on the final answer text rather than the derivation. However, the primary error is in the setup description: dividing by the constant term 4 yields r = 1 / (1 + 3/4 sin(theta)), so ed = 1, not ed = 4/4 as implied by the numerator manipulation in step 2 which is algebraically correct but the interpretation of 'ed' in step 4 is slightly garbled (it says ed=1, so d=4/3, which is correct, but the previous line showed 4/4 which is 1). The main issue is that the sentence in step 1 is too vague and doesn't guide the student to distinguish between sin (horizontal directrix) and cos (vertical directrix), which is crucial for identifying the directrix equation y=4/3 vs x=4/3.gpt-oss:20b: fail (misleading) 2026-10-16 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ); the given equation involves sinθ, so the directrix is horizontal (y = d).qwen3.6:27b-mlx: pass 2026-10-15gpt-oss:20b: fail (misleading) 2026-10-15 — The solution incorrectly states the directrix as y=4/3; for r=ed/(1+e sinθ) the directrix is y=−d, i.e. y=−4/3. The initial sentence also incorrectly references cosθ instead of sinθ, but the main error is the sign of the directrix.
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.