The length of the common chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the circle $x^2 + y^2 = 9$ lying in the first quadrant equals _____.
Step-by-Step Solution
Key Concept: Subtract ellipse from circle: $y^2(1 - 4/9) \cdot \dots$ — find intersection points. On both curves: from circle $y^2 = 9 - x^2$; into ellipse $x^2/9 + (9 - x^2)/4 = 1 \Rightarrow 4x^2 + 81 - 9x^2 = 36 \Rightarrow x^2 = 9$, $x = 3, y = 0$ (vertex). The common chord in the first quadrant is actually just the point $(3, 0)$; the curve also intersects at $(0, 2)...$ compute the chord length between these.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: $\sqrt{13}$