A plane passes through a fixed point (a, b, c). Show that the locus of the foot of the perpendic

step 1 Hello. Today we are doing a question which says a plane passes through a fixed point A B .C. Sho

A plane passes through a fixed point (a, b, c). Show that...

Key Concepts

Plane in Three-Dimensional Space

This concept refers to the representation and equation of a plane in 3D space, typically defined by a point through which the plane passes and a normal vector. The general equation n · (r – r?) = 0 encapsulates the idea that any point on the plane must yield a dot product of zero when offset by the known point, thereby encoding the plane’s orientation and position.

Foot of the Perpendicular

The foot of the perpendicular from a point to a plane is the unique point on the plane that is closest to the given point. It is found by projecting the point onto the plane along the line parallel to the normal vector of the plane, ensuring that the connecting line segment is orthogonal to the plane.

Geometric Locus

A locus is the collection of all points that satisfy a particular geometric condition. In analytic geometry, determining the locus involves finding an algebraic equation whose solutions represent all possible points that meet the defined criteria, such as being the foot of a perpendicular from a fixed point.

Orthogonal Projection

Orthogonal projection is the process of mapping a point onto a plane or line such that the line connecting the point and its projection is perpendicular to the plane or line. This concept is essential when computing the foot of the perpendicular, as it relies on properties of vector projection and the dot product.

Quadratic Surfaces

Quadratic surfaces are geometric objects defined by second-degree polynomial equations in three variables. In problems involving loci of points, the resulting equation often takes a quadratic form, representing surfaces like spheres or ellipsoids, which play a key role in the study of conic sections and quadric surfaces within analytic geometry.

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