Conics in polar form
Problem 8.318 · medium
For the conic \( \displaystyle r = \frac{8}{3 - 2 \cos{\left(\theta \right)}} \), 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).
- \[ \frac{8}{3 \left(1 - \frac{2 \cos{\left(\theta \right)}}{3}\right)} = \frac{8}{3 - 2 \cos{\left(\theta \right)}} \]The standard form.✓ Proved
- \[ \frac{\left|{-2}\right|}{3} = \frac{2}{3} \]e = 2/3: < 1, an ellipse.✓ Proved
- \[ \frac{8}{3 \cdot \frac{2}{3}} = 4 \]ed = 8/3, so d = 4.✓ Proved
Answer \( e = \frac{2}{3},\ \text{ellipse},\ \text{directrix } x = -4 \)
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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as x = -4, but the standard form derived is r = ed / (1 - e cos(theta)), which corresponds to a directrix at x = -d (i.e
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is identified as x = -4, but the standard form derived is r = ed / (1 - e cos(theta)), which corresponds to a directrix at x = -d (i.egpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — The directrix is identified as x = -4, but the standard form r = ed / (1 - e cos θ) corresponds to a directrix at x = -d. Since d = 4, the directrix is indeed x = -4. However, the solution fails to explicitly state the sign convention linking the denominator term (-2/3 cos θ) to the position of the directrix (x = -d vs x = d). While the final answer is correct, the reasoning in step 4 is incomplete and potentially misleading regarding how the sign in the denominator determines the directrix location. A student might incorrectly assume d is always positive and x = d, or miss the connection between the minus sign in the denominator and the negative x-intercept. More critically, step 1 says 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos θ)'. The resulting equation in step 2 is r = (8/3) / (1 - (2/3)cosθ). Here e = 2/3 and ed = 8/3, so d = 4. The form is 1 - e cos θ, which implies the directrix is x = -d = -4. The solution calculates d=4 but does not explain why the directrix is x=-4 rather than x=4. This is a significant gap in the logical derivation of the directrix's position.gpt-oss:20b: fail (misleading) 2026-10-15 — The solution claims the directrix is x = –4, but for the form r = ed/(1 – e cosθ) the directrix lies at x = +d. The sign of the directrix is reversed, giving a false result for the student.qwen3.6:27b-mlx: fail (error) 2026-10-14 — The directrix is incorrectly identified as x = -4. Since the term is -2 cos(theta), the directrix is vertical and located at x = -d, which is x = -4. Wait, let me re-evaluate. Standard form r = ed / (1 - e cos theta) corresponds to directrix x = -d. Here we have 1 - (2/3)cos(theta), so e=2/3, d=4. The directrix is x = -4. The stated answer says x = -4. The solution calculates d=4 but does not explicitly state the sign/direction logic for the directrix location, merely calculating d. However, looking closer at step 1, it 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. The solution fails to explicitly connect the sign in the denominator to the position of the directrix (x = -d vs x = d). While the final answer in the problem statement is correct, the solution text provided is incomplete because it calculates 'd' but doesn't state the directrix equation itself, leaving the student to guess the sign convention. Actually, looking at the prompt, the 'Stated answer' is separate from the 'Solution' lines. The solution lines end at step 4. Step 4 concludes 'd=4'. It does not conclude 'directrix x = -4'. Therefore, the solution is incomplete/incorrect because it doesn't derive the final answer requested (the directrix equation). It stops at finding d. A student might think the directrix is x=4. Thus, it is misleading/error.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.