Transforming a Sphere into an Ellipsoid?

📌 CONCEPT: When a sphere is transformed into an ellipsoid by replacing the x term with (x + z)^2, it results in a change in its shape and orientation in three-dimensional space.

📐 RULE / FORMULA: The new equation of the surface is obtained by replacing x^2 with (x + z)^2 in the original equation of the sphere. This results in the equation (x + z)^2 + y^2 + z^2 = 4.

💡 WORKED EXAMPLE: If we start with the equation of a sphere x^2 + y^2 + z^2 = 4, replacing x^2 with (x + z)^2 gives us the equation (x + z)^2 + y^2 + z^2 = 4. This simplifies to x^2 + 2xz + z^2 + y^2 + z^2 = 4, which further simplifies to x^2 + 2xz + y^2 + 2z^2 = 4. This represents the equation of an ellipsoid.

⚠️ COMMON MISTAKE: Students often fail to correctly replace the x term with (x + z)^2, leading to incorrect equations of the transformed surface.