Finding the Equation of a Plane Equidistant from Two Points with a Given Coefficient of x

In this article, we will explore the process of finding the equation of a plane that consists of all points equidistant from two given points and has a specific coefficient of x.

Step 1: Identify the Given Information

We are given the following information:

Two points: (5, 3, -4) and (3, -5, -2)
The coefficient of x in the equation of the plane is -2

Step 2: Determine the Equation of the Plane

To find the equation of the plane, we can use the following formula: Ax + By + Cz + D = 0

Where:

A, B, and C are the coefficients of x, y, and z, respectively.
D is the constant term.

We need to find the values of A, B, C, and D.

Step 2.1: Find the Vector Perpendicular to the Plane

The vector perpendicular to the plane can be found by taking the cross product of the vectors connecting the two given points.

Let's define the vectors:

Vector 1: (5, 3, -4) - (3, -5, -2) = (2, 8, -2)
Vector 2: (3, -5, -2) - (5, 3, -4) = (-2, -8, 2)

The cross product of these two vectors is:

Vector Perpendicular to the Plane = (2, 8, -2) × (-2, -8, 2) = (32, -4, -16)

Therefore, the coefficients of the plane equation are:

A = 32
B = -4
C = -16

Step 2.2: Determine the Constant Term (D)

We know that the coefficient of x is -2, so we can use this information to find the constant term (D).

The equation of the plane can be written as: 32x - 4y - 16z + D = 0

Rearranging the equation to isolate D, we get: D = 32(-2) - 4(3) - 16(-4) = -64 + 12 + 64 = 12

Therefore, the equation of the plane is: 32x - 4y - 16z + 12 = 0

Conclusion

In this article, we have demonstrated the step-by-step process of finding the equation of a plane that consists of all points equidistant from two given points and has a specific coefficient of x. By using the given information and applying the formula for the equation of a plane, we were able to determine the coefficients and the constant term, ultimately arriving at the final equation of the plane.