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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=(x4)2+(y0)2+(z0)2

= x2+168x+y2+z2

PB=(x+4)2+(y0)2+(z0)2

= x2+16+8x+y2+z2

It is given that PA+PB=10

    x2+168x+y2+z2

+x2+16+8x+y2+z2=10

        x2+168x+y2+z2

= 10x2+16+8x+y2+z2

Squaring both sides, we have

x2+168x+y2+z2

= 100+x2+16+8x+y2+z2

20x2+16+8x+y2+z2

        20x2+16+8x+y2+z2=16x+100

        5x2+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+25z216x2625200x=0

        9x2+25y2+25z2225=0

Thus the required equation is

9x2+25y2+25z2225=0.


Mathematics

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