The set of all points equidistant from (0, 1, 5) and 5, -1, 3) is a plane. Find the equation.
Let P(x, y, z) be a point equidistant from (0, 1, 5) and (5, -1, 3). The distance between P and (0, 1, 5) is equal to the distance between P and (5, -1, 3).
Using the distance formula, we can write the following equation:
√((x - 0)^2 + (y - 1)^2 + (z - 5)^2) = √((x - 5)^2 + (y + 1)^2 + (z - 3)^2)
Simplifying this equation, we get:
(x^2 + y^2 + z^2 - 10z + 26) = (x^2 - 10x + 25 + y^2 + 2y + 1 + z^2 - 6z + 9)
Canceling out the common terms, we get:
-10z + 26 = -10x + 25 + 2y + 1 - 6z + 9
Rearranging this equation, we get:
-10z + 10x + 6z - 2y = 9 - 25 - 1 - 9 + 26
Combining like terms, we have:
10x - 4z - 2y = 1
Dividing both sides by 2, we get:
5x - 2z - y = 1
Therefore, the equation of the plane is 5x - 2z - y = 1.
To find the equation of the plane that consists of all points equidistant from (0, 1, 5) and (5, -1, 3), we can use the formula for the distance between a point and a plane.
1. Find the midpoint between the given points:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
Midpoint = ((0 + 5)/2, (1 + -1)/2, (5 + 3)/2)
Midpoint = (5/2, 0, 4/2)
Midpoint = (5/2, 0, 2)
2. Calculate the distance between one of the given points and the midpoint:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Distance = sqrt((5 - 5/2)^2 + (-1 - 0)^2 + (3 - 2)^2)
Distance = sqrt((5/2)^2 + (-1)^2 + 1^2)
Distance = sqrt(25/4 + 1 + 1)
Distance = sqrt(25/4 + 4/4 + 4/4)
Distance = sqrt(33/4)
Distance = √33/2
3. The equation of the plane is of the form Ax + By + Cz + D = 0, where A, B, C, and D are constants to be determined.
4. Plug in the coordinates of the midpoint (5/2, 0, 2) and the distance (√33/2) into the equation:
A*(5/2) + B*(0) + C*(2) + D = (√33/2)
5. Simplify the equation:
(5/2)A + 2C + D = (√33/2)
6. To make things simpler, let's assume A = 2 to eliminate fractions:
(5/2)(2) + 2C + D = (√33/2)
5 + 2C + D = (√33/2)
7. Rearrange the equation:
2C + D = (√33/2) - 5
So, the equation of the plane that consists of all points equidistant from (0, 1, 5) and (5, -1, 3) is:
2C + D = (√33/2) - 5