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Conics in polar form

Problem 8.513 · medium

For the conic \( \displaystyle r = \frac{6}{4 - 7 \cos{\left(\theta \right)}} \), 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).
\[ \frac{3}{2 \left(1 - \frac{7 \cos{\left(\theta \right)}}{4}\right)} = \frac{6}{4 - 7 \cos{\left(\theta \right)}} \]
The standard form.✓ Proved
\[ \frac{\left|{-7}\right|}{4} = \frac{7}{4} \]
e = 7/4: > 1, a hyperbola.✓ Proved
\[ \frac{6}{4 \cdot \frac{7}{4}} = \frac{6}{7} \]
ed = 3/2, so d = 6/7.✓ Proved

Answer \( e = \frac{7}{4},\ \text{hyperbola},\ \text{directrix } x = -6/7 \)

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 solution incorrectly identifies the distance to the directrix. From r=(6/4)/(1-(7/4)cosθ) we have ed=6/7, so d=(ed)/e=(6/7)/(7/4)=24/49, not 6/7. Consequently the directrix x=−6/7 is wrong.
qwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as x = -6/7. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d, which should be x = -6/7? Wait, let's re-verify. Standard form r = ed / (1 + e cos theta) -> x = d. r = ed / (1 - e cos theta) -> x = -d. Here we have 1 - (7/4)cos(theta), so e=7/4, ed = 3/2 => d = 6/7. Since it is minus cos, the directrix is x = -d = -6/7. The stated answer says x = -6/7. The solution calculates d=6/7 but does not explicitly state the sign logic for the directrix location, merely calculating d. However, the stated answer in the prompt matches the calculation. Let me re-read carefully. 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 equation is r = 6 / (4 - 7 cos theta) = (6/4) / (1 - (7/4) cos theta). So e = 7/4, ed = 3/2 => d = 6/7. Since the term is - e cos theta, the directrix is x = -d = -6/7. The stated answer is: e = 7/4, hyperbola, directrix x = -6/7. The solution steps: 1. Divide to get standard form. 2. Shows algebra. 3. Identifies e. 4. Calculates d = 6/7. The solution fails to explicitly connect the sign in the denominator to the sign of the directrix coordinate. It calculates 'd' (the distance) but the final answer requires the coordinate. While the stated answer is correct, the solution text provided in the steps does not justify the negative sign for the directrix x-coordinate. It just says 'd = 6/7'. A student might think the directrix is x = 6/7. The step is incomplete/misleading regarding the final coordinate.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The directrix is incorrectly identified as x = -6/7. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d, which should be x = -6/7? Wait, let's re-verify. Standard form r = ed / (1 + e cos theta) -> x = d. r = ed / (1 - e cos theta) -> x = -d. Here we have 1 - (7/4)cos(theta), so e=7/4, ed = 3/2 => d = 6/7. Since it is minus cos, the directrix is x = -d = -6/7. The stated answer says x = -6/7. The solution calculates d=6/7 but does not explicitly state the sign logic for the directrix location, merely calculating d. However, the stated answer in the prompt matches the calculation. Let me re-read carefully. 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 equation is r = 6 / (4 - 7 cos theta) = (6/4) / (1 - (7/4) cos theta). So e = 7/4, ed = 3/2 => d = 6/7. Since the term is - e cos theta, the directrix is x = -d = -6/7. The stated answer is: e = 7/4, hyperbola, directrix x = -6/7. The solution steps: 1. Divide to get standard form. 2. Shows algebra. 3. Identifies e. 4. Calculates d = 6/7. The solution fails to explicitly connect the sign in the denominator to the sign of the directrix coordinate. It calculates 'd' (the distance) but the final answer requires the coordinate. While the stated answer is correct, the solution text provided in the steps does not justify the negative sign for the directrix x-coordinate. It just says 'd = 6/7'. A student might think the directrix is x = 6/7. The step is incomplete/misleading regarding the final coordinate.
gpt-oss:20b: fail (error) 2026-10-19 — The solution incorrectly identifies the distance to the directrix. From r=(6/4)/(1-(7/4)cosθ) we have ed=6/7, so d=(ed)/e=(6/7)/(7/4)=24/49, not 6/7. Consequently the directrix x=−6/7 is wrong.
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The directrix is incorrectly identified as x = -6/7. Since the standard form has a negative cosine term (1 - e cos θ), the directrix is vertical and located at x = -d, which is x = -6/7. Wait, let me re-evaluate. The standard form is r = ed / (1 - e cos θ). This corresponds to a directrix at x = -d. Here d = 6/7, so the directrix is x = -6/7. The stated answer says x = -6/7. The solution calculates d = 6/7 but does not explicitly state the sign of the directrix equation in the text, though the final answer in the problem statement matches. However, looking closely at step 4, it says 'ed = 3/2, so d = 6/7'. It fails to connect d to the specific directrix equation x = -d vs x = d. The standard form r = ed/(1 - e cos theta) implies the directrix is x = -d. The solution finds d but doesn't explicitly derive the equation x = -6/7, leaving a gap in logic for the student to know why it's negative. Actually, looking at the provided 'Stated answer', it is x = -6/7. The solution steps 1-4 do not produce the final equation. Step 4 just finds d. The connection between the sign in the denominator and the sign of the directrix is missing. This is a logical gap.
gpt-oss:20b: fail (error) 2026-10-18 — The solution incorrectly identifies the directrix. From r=6/(4-7cosθ)= (6/4)/(1-(7/4)cosθ), the denominator is 1‑e cosθ, which corresponds to a directrix at x=+d, not x=−d. Thus the statement that the directrix is x=−6/7 is false.
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as x = -6/7. For the form r = ed / (1 - e cos(theta)), the directrix is x = -d. Since d = 6/7, the directrix
gpt-oss:20b: pass 2026-10-18

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.