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Conics in polar form

Problem 8.545 · medium

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

Answer \( e = \frac{3}{4},\ \text{ellipse},\ \text{directrix } x = -7/3 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The directrix is incorrectly identified as x = -7/3. Since the standard form has a minus sign (1 - e cos θ), the directrix is at x = +d, so it should be x = 7/3.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-05 — The directrix is incorrectly identified as x = -7/3. Since the standard form has a minus sign (1 - e cos θ), the directrix is at x = +d, so it should be x = 7/3.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: fail (error) 2026-11-04 — The directrix is incorrectly identified as x = -7/3. Since the standard form has a negative cosine term (1 - e cos theta), the directrix is vertical and located at x = -d, which is x = -7/3. Wait, let me re-evaluate. The standard form is r = ed / (1 - e cos theta). This corresponds to a directrix at x = -d. Here d = 7/3, so x = -7/3. The stated answer says x = -7/3. The solution calculates d = 7/3 but does not explicitly state the sign or position of the directrix in the sentences, only calculating d. However, looking at the stated answer provided in the prompt: 'directrix x = -7/3'. The solution steps calculate d=7/3. Step 4 says 'ed = 7/4, so d = 7/3'. It does not explicitly conclude the equation of the directrix. But usually, 'find the directrix' implies giving the equation. The solution stops at finding d. Is this an error? The prompt asks to review the *solution*. The solution lines are 1-4. Line 4 finds d. It does not state the final answer for the directrix equation. However, the 'Stated answer' is separate. The solution itself is incomplete if it doesn't state the directrix equation. But more importantly, let's look at the sign. r = 7/(4-3cos(theta)) = (7/4)/(1 - (3/4)cos(theta)). This is of the form ed/(1 - e cos theta). This form corresponds to a directrix at x = -d. So x = -7/3. The stated answer is correct. The solution calculates d=7/3. It fails to explicitly state the directrix is x = -7/3, but perhaps that's implied? No, 'find the directrix' requires the equation. The solution is incomplete. Also, Step 1 says 'Divide top and bottom... to get r = ed/(1 +/- e cos theta)'. Step 2 shows the algebra. Step 3 finds e. Step 4 finds d. It never connects d to the directrix equation x = -d. This is a missing step/conclusion. Is it an error? It's an incomplete solution. But is it 'wrong'? It doesn't teach something false, it just doesn't finish. However, often in these reviews, if the final answer isn't derived, it's a fail. Let's look closer. The prompt says 'Stated answer: ... directrix x = -7/3'. The solution does not produce this statement. It produces d=7/3. A student might think the directrix is x = 7/3 if they don't know the sign convention. The solution does not clarify the sign convention for the directrix based on the minus sign in the denominator. This is misleading/incomplete.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: inconclusive 2026-11-04 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as x = -7/3. Since the standard form has a positive cosine term (after factoring out the negative sign, the
gpt-oss:20b: pass 2026-11-04

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 2026-11-05 with SymPy 1.14.0.