Find the equation of set of points P such that PA2+PB2=2k2, where A and B are the points (3,4,5) and (−1,3,−7), respectively.
Get the complete, step-by-step math solution for: "Find the equation of set of points {P} such that {PA}^{2}+ {PB}^{2}=2 k^{2}, where {A} and {B} are the points (3{,}4{,}5) and (-1{,}3,-7), respectivel...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define coordinates of points
We start by defining the coordinates of the unknown point P as (x, y, z) and the given points A and B with their respective coordinates.
Step 2: Calculate squared distances PA² and PB²
Using the distance formula, the square of the distance between two points (x1,y1,z1) and (x2,y2,z2) is (x2−x1)2+(y2−y1)2+(z2−z1)2. We apply this formula to find PA2 and PB2.
Step 3: Substitute into the given equation
Now, we substitute the expressions for PA2 and PB2 into the given equation PA2+PB2=2k2.
Step 4: Expand and simplify the equation
We expand all the squared terms and then combine like terms (x2, x, y2, y, z2, z, and constant terms) to simplify the equation.
Step 5: Divide by 2 and rearrange
To further simplify, we divide the entire equation by 2. Then, we move the k2 term to the left side to express the equation in a standard form.