a sphere of constant radius passes trough origin and cuts axis in A,B,C. find locus of foot of perpendicular from 0 to plane ABC

To solve the problem of finding the locus of the foot of the perpendicular from the origin to the plane defined by points A, B, and C, we first need to understand the geometric setup involving the sphere and the plane.

Understanding the Sphere and Points A, B, C

Let's denote the center of the sphere as O and its radius as r. Given that the sphere passes through the origin, we can express its equation as:

x² + y² + z² = r².

Points A, B, and C are points where the sphere intersects the coordinate axes. For instance, if the sphere intersects the x-axis at point A, the coordinates could be (r, 0, 0). Similarly, points B and C could be represented as (0, r, 0) and (0, 0, r), respectively.

Finding the Plane Equation

The coordinates of points A, B, and C lead us to define the plane that contains these points. The general equation of a plane through three points can be derived using the determinant method. In this case, the points can be represented as:

• A (r, 0, 0)
• B (0, r, 0)
• C (0, 0, r)

The equation of the plane can be formulated as follows:

(x - r)(y - r)(z - r) = 0,

which simplifies to:

x + y + z = r.

Foot of the Perpendicular from the Origin

Next, we need to find the foot of the perpendicular from the origin (0, 0, 0) to this plane. The equation of the plane can be manipulated to find the coordinates of the foot of the perpendicular. The foot of the perpendicular can be found using the formula:

P = k(1, 1, 1),

where k is a scalar that adjusts the distance from the origin to the plane. Since the point P lies on the plane, we substitute it into the plane equation:

k + k + k = r,

which simplifies to:

3k = r.

Thus, we find:

k = r/3.

Coordinates of the Foot of the Perpendicular

Substituting k back into our expression for P gives:

P = (r/3, r/3, r/3).

This means that the foot of the perpendicular from the origin to the plane ABC is located at (r/3, r/3, r/3).

Determining the Locus

As the radius r of the sphere changes, the coordinates of the foot of the perpendicular will trace out a line in space. Specifically, as r varies, the locus of the foot of the perpendicular can be expressed parametrically as:

(x, y, z) = (t, t, t),

where t = r/3. Therefore, the locus of points is along the line defined by x = y = z.

Final Result

In conclusion, the locus of the foot of the perpendicular from the origin to the plane ABC, as the sphere's radius varies, is the line described by:

x = y = z.

This is a diagonal line in three-dimensional space that passes through the origin and extends infinitely along the direction where all coordinates are equal. Thus, any point on this line is the foot of the perpendicular from the origin to the plane formed by points A, B, and C.