Find an equation of the sphere with center (2,-6,4) and radius 5 . Describe its intersection wit

step 1 So suppose we have a sphere with the center at 2, negative 6, 4, and a radius of 5. What would i

Find an equation of the sphere with center (2,-6,4) and...

Key Concepts

Distance from a Point to a Plane

This concept is used to calculate the shortest distance between a given point (such as the center of a sphere) and a specified plane (such as one of the coordinate planes). The computed distance is essential to determine the radius of the circle formed by the intersection of the sphere with the plane, as it helps establish the relationship between the sphere’s radius and the size of the cross-sectional circle.

Intersection of a Sphere with a Plane

When a plane intersects a sphere, the cross-section is typically a circle. The nature of this intersection depends on the distance from the sphere's center to the intersecting plane. If the distance is less than the radius of the sphere, the intersection is a circle whose radius can be determined using the Pythagorean theorem, linking the sphere's radius, the distance from the center to the plane, and the circle's radius in the cross-sectional view.

Coordinate Planes

Coordinate planes are the three fundamental planes in three-dimensional space defined by setting one of the coordinates to zero (x=0, y=0, or z=0). These planes serve as reference frameworks for analyzing geometric figures in 3D, and studying intersections with these planes often simplifies the understanding of spatial relationships.

Sphere Equation

This concept involves expressing the set of all points that are equidistant from a fixed center point in space. The standard form (x - h)² + (y - k)² + (z - l)² = r² encapsulates the idea that every point (x, y, z) on the sphere is at a distance r from the center (h, k, l). This equation is fundamental in geometry and calculus for describing spherical surfaces.

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