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Conics in polar form

Problem 8.313 · medium

For the conic \( \displaystyle r = \frac{11}{5 - 7 \cos{\left(\theta \right)}} \), 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).
\[ \frac{11}{5 \left(1 - \frac{7 \cos{\left(\theta \right)}}{5}\right)} = \frac{11}{5 - 7 \cos{\left(\theta \right)}} \]
The standard form.✓ Proved
\[ \frac{\left|{-7}\right|}{5} = \frac{7}{5} \]
e = 7/5: > 1, a hyperbola.✓ Proved
\[ \frac{11}{5 \cdot \frac{7}{5}} = \frac{11}{7} \]
ed = 11/5, so d = 11/7.✓ Proved

Answer \( e = \frac{7}{5},\ \text{hyperbola},\ \text{directrix } x = -11/7 \)

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 directrix is incorrectly given as x = -11/7; for the standard form r = ed/(1 – e cosθ) the directrix is x = +d = 11/7.
qwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as x = -11/7. Since the standard form derived is r = ed / (1 + e cos(theta)) (with a negative cosine term in the original denominator becoming positive in the standard form denominator after factoring out -7/5, wait: 5 - 7cos = 5(1 - 7/5 cos). The form is 1 - e cos theta. This corresponds to a directrix at x = -d. So x = -11/7 is actually correct for the form 1 - e cos theta? Let's re-evaluate. Standard forms: r = ed/(1+e cos theta) -> x=d. r = ed/(1-e cos theta) -> x=-d. The equation is r = (11/5) / (1 - (7/5)cos theta). Here e=7/5, ed=11/5 => d=11/7. The form is 1 - e cos theta, which corresponds to directrix x = -d = -11/7. The stated answer says x = -11/7. The solution calculates d=11/7 but does not explicitly state the sign logic for the directrix location, just calculating 'd'. However, the final answer in the problem statement matches the calculation. Is there an error? Let's check the sign carefully. r = 11 / (5 - 7 cos theta). Divide by 5: r = (11/5) / (1 - (7/5) cos theta). This is of the form r = ed / (1 - e cos theta). For this form, the directrix is x = -d. d = (11/5)/(7/5) = 11/7. So directrix is x = -11/7. The solution finds d=11/7. It does not explicitly conclude x=-11/7 in the steps, but the stated answer has it. The prompt asks to review the solution lines. Line 4 calculates d. It doesn't state the directrix equation. The stated answer is separate. The solution steps are technically correct in finding e and d. But does it identify the directrix? The problem asks to find the directrix. The solution stops at finding d. It misses the step of mapping d to the coordinate equation x = -d. This is an incomplete solution, arguably an error in fulfilling the prompt's request to 'find the directrix'. Also, the sign of the cosine term determines the side. The solution ignores the sign implication for the directrix location in the text. It just computes d. A student might think d is the directrix value directly. This is misleading/incomplete.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The directrix is incorrectly identified as x = -11/7. Since the standard form derived is r = ed / (1 + e cos(theta)) (with a negative cosine term in the original denominator becoming positive in the standard form denominator after factoring out -7/5, wait: 5 - 7cos = 5(1 - 7/5 cos). The form is 1 - e cos theta. This corresponds to a directrix at x = -d. So x = -11/7 is actually correct for the form 1 - e cos theta? Let's re-evaluate. Standard forms: r = ed/(1+e cos theta) -> x=d. r = ed/(1-e cos theta) -> x=-d. The equation is r = (11/5) / (1 - (7/5)cos theta). Here e=7/5, ed=11/5 => d=11/7. The form is 1 - e cos theta, which corresponds to directrix x = -d = -11/7. The stated answer says x = -11/7. The solution calculates d=11/7 but does not explicitly state the sign logic for the directrix location, just calculating 'd'. However, the final answer in the problem statement matches the calculation. Is there an error? Let's check the sign carefully. r = 11 / (5 - 7 cos theta). Divide by 5: r = (11/5) / (1 - (7/5) cos theta). This is of the form r = ed / (1 - e cos theta). For this form, the directrix is x = -d. d = (11/5)/(7/5) = 11/7. So directrix is x = -11/7. The solution finds d=11/7. It does not explicitly conclude x=-11/7 in the steps, but the stated answer has it. The prompt asks to review the solution lines. Line 4 calculates d. It doesn't state the directrix equation. The stated answer is separate. The solution steps are technically correct in finding e and d. But does it identify the directrix? The problem asks to find the directrix. The solution stops at finding d. It misses the step of mapping d to the coordinate equation x = -d. This is an incomplete solution, arguably an error in fulfilling the prompt's request to 'find the directrix'. Also, the sign of the cosine term determines the side. The solution ignores the sign implication for the directrix location in the text. It just computes d. A student might think d is the directrix value directly. This is misleading/incomplete.
gpt-oss:20b: fail (error) 2026-10-17 — The directrix is incorrectly given as x = -11/7; for the standard form r = ed/(1 – e cosθ) the directrix is x = +d = 11/7.
qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The directrix is identified as x = -11/7, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = +d (x = 11/7). The negative sign in the denominator indicates the directrix is to the right of the pole, not the left.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (style) 2026-10-14 — [domain objection, downgraded to style] The directrix is identified as x = -11/7, but the standard form r = ed/(1 - e cos θ) corresponds to a directrix at x = +d (i.e., x = 11/7). The negative sign in the denominator indicates the directrix is to the right of the pole, not the left.
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.