Conics in polar form
Problem 8.457 · medium
For the conic \( \displaystyle r = \frac{10}{6 \sin{\left(\theta \right)} + 5} \), 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).
- \[ \frac{2}{\frac{6 \sin{\left(\theta \right)}}{5} + 1} = \frac{10}{6 \sin{\left(\theta \right)} + 5} \]The standard form.✓ Proved
- \[ \frac{\left|{6}\right|}{5} = \frac{6}{5} \]e = 6/5: > 1, a hyperbola.✓ Proved
- \[ \frac{10}{5 \cdot \frac{6}{5}} = \frac{5}{3} \]ed = 2, so d = 5/3.✓ Proved
Answer \( e = \frac{6}{5},\ \text{hyperbola},\ \text{directrix } y = 5/3 \)
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: fail (error) — The first sentence incorrectly states the standard polar form as r=ed/(1±e cosθ) and claims to obtain it by dividing by the constant term. The given conic involves sinθ, not cosθ, so the form should be r=ed/(1±e sinθ). This misstatement misleads the reader about the correct orientation and form of the equation.qwen3.6:27b-mlx: fail (error) — The solution fails to account for the sine term in the denominator, which implies the directrix is horizontal (y = d), not vertical. While the calculated value 5/3 is correct for d, the final answer 'y = 5/3' is only correct if the sign is handled properly; however, the standard form derivation in step 1 is ambiguous regarding the sign of the directrix. More critically, the standard form r = ed / (1 + e sin theta) corresponds to a directrix y = +d. The given equation is r = 2 / (1 + 1.2 sin theta). Here ed = 2 and e = 1.2, so d = 2/1.2 = 5/3. Since the sign in the denominator is positive (+ sin theta), the directrix is y = +5/3. The solution is actually correct in its final result, but the reasoning in step 1 is vague ('±') and step 4 jumps to 'ed=2' without explicitly showing the numerator manipulation clearly enough for a student to follow the sign convention. Wait, let's re-read carefully. Step 2 shows (10/5) / ((6/5)sin + 1). This is 2 / (1 + 1.2 sin). This matches r = ed / (1 + e sin theta). So ed = 2, e = 6/5. d = 2 / (6/5) = 10/6 = 5/3. Since it is + sin, directrix is y = d = 5/3. The solution is correct. Why did I think it was wrong? Let me check the 'adversarial' instruction. Is there a subtle error? Let's look at Step 1: 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos theta)'. It says 'cos theta' but the problem has 'sin theta'. This is a misleading statement in the sentence. It claims the target form involves cos, but the problem involves sin. This is a 'misleading' or 'error' in the sentence description. The student might think they need to convert sin to cos or that the formula is only for cos. The sentence is factually incorrect regarding the general standard forms (which include sin). Also, Step 4 says 'ed = 2, so d = 5/3'. It doesn't explicitly state the directrix equation is y = 5/3, it just calculates d. The final answer is stated in the problem description, not derived in the last sentence. The last sentence just calculates d. The connection between d and the directrix equation (y=d vs y=-d vs x=d) is missing from the sentences. Step 1 mentions cos, implying x-directrix, but the problem is sin, implying y-directrix. This is a significant conceptual gap/error in the explanation.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution fails to account for the sine term in the denominator, which implies the directrix is horizontal (y = d), not vertical. While the calculated value 5/3 is correct for d, the final answer 'y = 5/3' is only correct if the sign is handled properly; however, the standard form derivation in step 1 is ambiguous regarding the sign of the directrix. More critically, the standard form r = ed / (1 + e sin theta) corresponds to a directrix y = +d. The given equation is r = 2 / (1 + 1.2 sin theta). Here ed = 2 and e = 1.2, so d = 2/1.2 = 5/3. Since the sign in the denominator is positive (+ sin theta), the directrix is y = +5/3. The solution is actually correct in its final result, but the reasoning in step 1 is vague ('±') and step 4 jumps to 'ed=2' without explicitly showing the numerator manipulation clearly enough for a student to follow the sign convention. Wait, let's re-read carefully. Step 2 shows (10/5) / ((6/5)sin + 1). This is 2 / (1 + 1.2 sin). This matches r = ed / (1 + e sin theta). So ed = 2, e = 6/5. d = 2 / (6/5) = 10/6 = 5/3. Since it is + sin, directrix is y = d = 5/3. The solution is correct. Why did I think it was wrong? Let me check the 'adversarial' instruction. Is there a subtle error? Let's look at Step 1: 'Divide top and bottom by the constant... to get r = ed/(1 ± e cos theta)'. It says 'cos theta' but the problem has 'sin theta'. This is a misleading statement in the sentence. It claims the target form involves cos, but the problem involves sin. This is a 'misleading' or 'error' in the sentence description. The student might think they need to convert sin to cos or that the formula is only for cos. The sentence is factually incorrect regarding the general standard forms (which include sin). Also, Step 4 says 'ed = 2, so d = 5/3'. It doesn't explicitly state the directrix equation is y = 5/3, it just calculates d. The final answer is stated in the problem description, not derived in the last sentence. The last sentence just calculates d. The connection between d and the directrix equation (y=d vs y=-d vs x=d) is missing from the sentences. Step 1 mentions cos, implying x-directrix, but the problem is sin, implying y-directrix. This is a significant conceptual gap/error in the explanation.gpt-oss:20b: fail (error) 2026-10-19 — The first sentence incorrectly states the standard polar form as r=ed/(1±e cosθ) and claims to obtain it by dividing by the constant term. The given conic involves sinθ, not cosθ, so the form should be r=ed/(1±e sinθ). This misstatement misleads the reader about the correct orientation and form of the equation.qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The directrix is incorrectly identified as y = 5/3. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but the positigpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (style) 2026-10-15 — [domain objection, downgraded to style] The solution incorrectly identifies the directrix as y = 5/3. Since the standard form involves sin(theta), the directrix is horizontal (y = d), but the sign in the denominator is positive (+ e sin(theta)), which corresponds to a directrix below the pole (y = -d). Thus, the directrix is y = -5/3.gpt-oss:20b: fail (error) 2026-10-15 — The first sentence incorrectly states the standard polar form as r = ed/(1 ± e cosθ); the given conic involves sinθ, so the correct form is r = ed/(1 ± e sinθ). This misstatement invalidates the subsequent reasoning.
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.