Arbitrary Constants in the Equation of a Plane

In mathematics, the equation of a plane is typically given in the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of x, y, and z respectively, and D is a constant. The coefficients A, B, and C determine the direction of the plane's normal vector, while the constant D determines the distance of the plane from the origin.

To determine the number of arbitrary constants in the equation of a plane, we need to consider the number of independent variables or parameters involved.

Variables and Parameters

In the equation of a plane, the variables are x, y, and z, which represent the coordinates of any point on the plane. These variables can take any real values.

On the other hand, the parameters or constants A, B, C, and D are fixed values that determine the characteristics of the plane. They are not variables and cannot take arbitrary values.

Degrees of Freedom

The degrees of freedom of a system refer to the number of independent variables or parameters that can be chosen freely. In the case of a plane, the degrees of freedom represent the number of parameters that can be varied without affecting the plane's essential characteristics.

In the equation of a plane, there are four parameters: A, B, C, and D. However, these parameters are not all independent of each other. The equation is homogeneous, meaning that if we scale all the parameters by a constant factor, the equation remains the same. This implies that three out of the four parameters are dependent.

Equation with Arbitrary Constants

We can rewrite the equation of a plane as follows:

Ax + By + Cz + D = 0
Dividing the equation by D, we get:

(A/D)x + (B/D)y + (C/D)z + 1 = 0
Letting p = A/D, q = B/D, and r = C/D, the equation becomes:

px + qy + rz + 1 = 0
In this form, the equation of the plane has three arbitrary constants: p, q, and r. These constants can take any real values, and they determine the direction and distance of the plane.

Therefore, the correct answer is option 'B' - there are three arbitrary constants in the equation of a plane.