Word Count : 349

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We are tasked with finding the equation of a sphere that: 1. Touches the plane \(8x + 5y + 3z + 1 = 0\) at the point \((3, -1, -1)\). 2. Cuts the sphere \(x^2 + y^2 + z^2 - 2x + y - z - 6 = 0\) orthogonally. ### Step 1: Equation of the Sphere The general equation of a sphere is given by: \[ (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2 \] where \((a, b, c)\) is the ________ ______ ___ ____ ______ ______ ______ _________ ______.
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