Conic Sections

We are given: - Ellipse: - A confocal hyperbola whose transverse axis is 2. Step 1: Standard Form of the Ellipse Dividing the given ellipse equation by 36: This is an ellipse with: - Semi-major axis - Semi-minor axis The focal distance is calculated using the relation: Step 2: Standard Form of the Hyperbola Since the given hyperbola is confocal with the ellipse, it must have the same focal distance . The standard form of a hyperbola with this condition is: We know the transverse axis is 2, so . Now using the identity for the focal distance in a hyperbola: Since and , Squaring both sides: Thus, the hyperbola is: Step 3: Finding the Points of Intersection At points of intersection, add the two equations: Simplifying, Finding a common denominator: From the ellipse equation: Substituting : Step 4: Equation of the Circle Now compute : Step 5: Final Answer