Problem

Find the equation of variation for a point P (x, y, z) that moves so that it is always equidistant from the points A(1, 2, 3) and B(4, 6, 8)

Answer

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Answer

Expanding and simplifying the equation, we get 3x+4y+6z=21.

Steps

Step 1 ：Given a point P(x, y, z) equidistant from points A(1, 2, 3) and B(4, 6, 8), the distance from P to A is equal to the distance from P to B. We can write this as AP=BP.

Step 2 ：The distance between two points in 3D space is given by the formula D=(x2−x1)2+(y2−y1)2+(z2−z1)2.

Step 3 ：Substituting the given points into the distance formula, we get (x−1)2+(y−2)2+(z−3)2=(x−4)2+(y−6)2+(z−8)2

Step 4 ：Squaring both sides to eliminate the square root, we get (x−1)2+(y−2)2+(z−3)2=(x−4)2+(y−6)2+(z−8)2

Step 5 ：Expanding and simplifying the equation, we get 3x+4y+6z=21.

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