Question 17

Intersection: x² + 4y² = 1 and 4x² + y² = 1 ⇒ x² = y² = 1/5, so P = (1/√5, 1/√5). Slopes of tangents: - For x² + 4y² = 1: dy/dx = −x/(4y) = −1/4. - For 4x² + y² = 1: dy/dx = −4x/y = −4. tan θ = |(m₁ − m₂)/(1 + m₁m₂)| = |(−1/4 + 4)/(1 + 1)| = (15/4)/2 = 15/8. So 4 tan θ = 15/2 = 7.5.

**Calculating intersections** I need to solve this problem straightforwardly. The intersections involve ellipses where x=y=1/sqrt5 with certain slopes. I’m calculating the acute angle differences, especially with the formula involving tangent. After some calculations, I find that I should keep the output concise, focusing solely on the JSON result, since that’s what the user requested. So, let’s finalize that as: {"answer":7.5,"unanswered":false}.

**Solving geometry problem** I need to tackle the question regarding the intersections of the ellipses defined by the equations x² + 4y² = 1 and 4x² + y² = 1. Focusing on the first quadrant, I find x = y. I calculate the intersection and slopes of the tangents. The acute angle between the tangents leads to determining values through the tangent formula, finally finding that 4 tan θ equals 7.5. I'll format this result into the required JSON structure. **Verifying angle calculations** Before finalizing, I want to double-check that there's no trick involved here. The angles formed by the tangents at point P have slopes of -1/4 and -4, leading to arctan calculations resulting in negative angles with the x-axis. The difference between those angles confirms the acute angle is around 62°, and 4 times the tangent results in 7.5. By structuring it in JSON, I ensure that the output is clean and includes nothing beyond the numerical answer.

Solving the two ellipse equations together, adding and subtracting gives x²+y²=2/5 and x²=y², so both equal 1/5, giving intersection point P=(1/√5, 1/√5). Now I need to find the tangent line to x²+4y²=1 at that point. Differentiating implicitly gives slope1=-1/4, and doing the same for 4x²+y²=1 gives slope2=-4. Using the angle-between-lines formula, tanθ = |(m1-m2)/(1+m1m2)| = (15/4)/2 = 15/8, so 4tanθ = 15/2 = 7.5. **Answer: 7.5**