Question

# Find the equation of the set of points P, the sum of whose distance from A(4, 0, 0) and B(−4, 0, 0) is equal to 10.

Solution

## Let P(x, y, z) be any point. Then PA=√(x−4)2+(y−0)2+(z−0)2 = √x2+16−8x+y2+z2 PB=√(x+4)2+(y−0)2+(z−0)2 = √x2+16+8x+y2+z2 It is given that PA+PB=10 ∴    √x2+16−8x+y2+z2 +√x2+16+8x+y2+z2=10 ⇒        √x2+16−8x+y2+z2 = 10−√x2+16+8x+y2+z2 Squaring both sides, we have x2+16−8x+y2+z2 = 100+x2+16+8x+y2+z2 −20√x2+16+8x+y2+z2 ⇒        20√x2+16+8x+y2+z2=16x+100 ⇒        5√x2+16+8x+y2+z2=4x+25 Squaring both sides again, we have 25(x2+16+8x+y2+z2)=16x2+625+200x ⇒        25x2+400+200x+25y2+25z2−16x2−625−200x=0 ⇒        9x2+25y2+25z2−225=0 Thus the required equation is 9x2+25y2+25z2−225=0. Mathematics

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