Home›Calculus 2›Conics in polar form›Problem 8.509

Conics in polar form

Problem 8.509 · medium

For the conic \( \displaystyle r = \frac{12}{3 \cos{\left(\theta \right)} + 4} \), 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).
Reviewed
\[ \frac{3}{\frac{3 \cos{\left(\theta \right)}}{4} + 1} = \frac{12}{3 \cos{\left(\theta \right)} + 4} \]
The standard form.✓ Proved
\[ \frac{\left|{3}\right|}{4} = \frac{3}{4} \]
e = 3/4: < 1, an ellipse.✓ Proved
\[ \frac{12}{4 \cdot \frac{3}{4}} = 4 \]
ed = 3, so d = 4.✓ Proved

Answer \( e = \frac{3}{4},\ \text{ellipse},\ \text{directrix } x = 4 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly transforms the polar equation into standard form, identifies the eccentricity and conic type, and accurately calculates the directrix distance.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly transforms the polar equation into standard form, identifies the eccentricity and conic type, and accurately calculates the directrix distance.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly manipulates the polar equation into standard form, identifies the eccentricity and conic type, and accurately calculates the directrix distance.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The calculation of d is incorrect. From ed = 3 and e = 3/4, d should be 4, but the equation shown (12/4)/(3/4) = 4 evaluates to 4, which is correct
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.