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Conics in polar form

Problem 8.442 · medium

For the conic \( \displaystyle r = \frac{8}{6 \cos{\left(\theta \right)} + 1} \), 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).
\[ \frac{8}{1 \left(\frac{6 \cos{\left(\theta \right)}}{1} + 1\right)} = \frac{8}{6 \cos{\left(\theta \right)} + 1} \]
The standard form.✓ Proved
\[ \frac{\left|{6}\right|}{1} = 6 \]
e = 6: > 1, a hyperbola.✓ Proved
\[ \frac{8}{1 \cdot 6} = \frac{4}{3} \]
ed = 8, so d = 4/3.✓ Proved

Answer \( e = 6,\ \text{hyperbola},\ \text{directrix } x = 4/3 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly divide the numerator and denominator by 6 to convert the equation into the standard form r = ed / (1 + e cos(theta)). Consequently, it incorrectly identifies the numerator 8 as 'ed' instead of the correct value 8/6, leading to an incorrect directrix calculation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to explicitly divide the numerator and denominator by 6 to convert the equation into the standard form r = ed / (1 + e cos(theta)). Consequently, it incorrectly identifies the numerator 8 as 'ed' instead of the correct value 8/6, leading to an incorrect directrix calculation.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to explicitly state the equation of the directrix (e.g., x = 4/3), only calculating the distance d. Additionally, the standard form derivation is presented as an identity rather than a transformation, obscuring the necessary algebraic step of dividing by 6.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to divide the denominator by the constant term (1) to normalize the equation into the standard form r = ed / (1 + e cos(theta)). As written, the denominator is 6 cos(theta) + 1, so the eccentricity is 6, but the numerator must also be divided by 1 (which is trivial) to match the form. However, the critical error is in step 4: the standard form is r = (ed) / (1 + e cos(theta)). The given equation is r = 8 / (1 + 6 cos(theta)). Thus, ed = 8 and e = 6, so d = 8/6 = 4/3. Wait, let's re-read carefully. The standard form requires the constant term in the denominator to be 1. The given denominator is 6 cos(theta) + 1. This is already in the form 1 + 6 cos(theta). So e = 6. Then ed = 8 => 6d = 8 => d = 4/3. The directrix is x = d = 4/3. The steps seem algebraically correct. Let me double check the 'unchecked' equations. Step 2 says (8/1)/((6/1)*cos(theta) + 1) = 8/(6*cos(theta) + 1). This is just rewriting. Step 3 says Abs(6)/1 = 6. This identifies e. Step 4 says (8/1)/6 = 4/3. This calculates d. The logic holds. Why did I think it was wrong? Ah, usually students miss dividing by the constant if it's not 1. Here the constant IS 1. So the setup is correct. The answer is correct. The sentences are minimal but correct. Wait, is the directrix x = 4/3 or x = -4/3? The form is 1 + e cos(theta), which corresponds to a directrix at x = d. Since e=6 and d=4/3, the directrix is x = 4/3. The solution is correct.
gpt-oss:20b: pass 2026-10-15

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.