Find the equation of the set of the points P such that its distances from the points A(3,4,−5) and B(−2,1,4) are equal.
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Step-by-Step Solution
Step 1: Define point P and the distance formula
Let the coordinates of point P be (x,y,z). The problem states that the distance from P to A is equal to the distance from P to B. We will use the standard 3D distance formula to express these distances.
Step 2: Set up the equality of distances
We set the distance PA equal to the distance PB. Substitute the coordinates of A (3,4,−5) and B (−2,1,4) into the distance formula. Remember that z−(−5) becomes z+5 and x−(−2) becomes x+2.
Step 3: Square both sides and expand
To eliminate the square roots, we square both sides of the equation. Then, we expand each squared term using the algebraic identity (a−b)2=a2−2ab+b2 and (a+b)2=a2+2ab+b2.
Step 4: Simplify the equation
Notice that x2, y2, and z2 terms cancel out from both sides of the equation. Combine the constant terms on each side. Then, move all terms to one side to simplify the equation.
Step 5: Rearrange to standard form
Gather all x, y, and z terms, and constant terms on one side of the equation. This gives us the final linear equation representing the set of points P.