Final answer:
To locate a plane equidistant from points S and P, find the midpoint M of SP and the vector SP. Use these to create the perpendicular bisector, which is the desired plane equation 3x - 5y + 9z = D.
Step-by-step explanation:
To find an equation for the set of all points equidistant from the points S=(−1,2,−5) and P=(2,−3,4), we need to find the perpendicular bisector of the line segment joining S and P. This set of points will form a plane in three-dimensional space. First, find the midpoint M of segment SP, which will lie on this plane. Then, determine the vector SP, which will be perpendicular to the plane.
The midpoint M is given by M = ((Sx+Px)/2, (Sy+Py)/2, (Sz+Pz)/2) = ((-1+2)/2, (2+(-3))/2, (-5+4)/2) = (0.5, -0.5, -0.5).
The vector SP is P - S = (2 - (-1), -3 - 2, 4 - (-5)) = (3, -5, 9).
The equation of the plane can be written in the form Ax + By + Cz = D, where A, B, and C are the components of the vector SP, and D can be found by substituting the coordinates of M into the equation: 3*0.5 -5*(-0.5) + 9*(-0.5) = D.
So, the equation of the plane is 3x - 5y + 9z = D, where D is the calculated value after substituting the midpoint's coordinates.