Question:
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Updated On: Oct 23, 2023
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Solution and Explanation
Let P (x, y, z) be the point that is equidistant from points A(1, 2, 3) and B(3, 2, -1).
Accordingly, PA = PB
⇒ PA2 = PB2
⇒(x-1)2+(y-2)2 +(z−3)2 = (x−3)2+(y−2)2 +(z+1)2
⇒ x2 - 2x + 1+ y2 - 4y+4+z2 - 6z + 9 = x2 - 6x+9+ y2 - 4y+ 4 + z2+2z+1
⇒ - 2x - 4y - 6z +14= - 6x - 4y+ 2z+14
⇒ - 2x - 6z + 6x - 2z = 0
⇒ 4x - 8z = 0
⇒x - 2z= 0
Thus, the required equation is x - 2z = 0.
Accordingly, PA = PB
⇒ PA2 = PB2
⇒(x-1)2+(y-2)2 +(z−3)2 = (x−3)2+(y−2)2 +(z+1)2
⇒ x2 - 2x + 1+ y2 - 4y+4+z2 - 6z + 9 = x2 - 6x+9+ y2 - 4y+ 4 + z2+2z+1
⇒ - 2x - 4y - 6z +14= - 6x - 4y+ 2z+14
⇒ - 2x - 6z + 6x - 2z = 0
⇒ 4x - 8z = 0
⇒x - 2z= 0
Thus, the required equation is x - 2z = 0.
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