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Question

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

Solution
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Let the coordinates of P be (x,y,z).

The coordinates of points A and B are (4,0,0) and (4,0,0) respectively.

It is given that PA+PB=10

(x4)2+y2+z2+(x+4)2+y2+z2=10

(x4)2+y2+z2=10(x+4)2+y2+z2

On squaring both sides, we obtain

(x4)2+y2+z2=100+(x+4)2+y2+z220(x+4)2+y2+z2

x28x+16+y2+z2=10020x2+8x+16+y2+z2+x2+8x+16+y2+z2

20x2+8x+16+y2+z2=100+16x

5x2+8x+16+y2+z2=(25+4x)

On squaring both sides again, we obtain

25(x2+8x+16+y2+z2)=625+16x2+200x

25x2+200x+400+25y2+25z2=625+16x2+200x

9x2+25y2+25z2225=0

Thus the required equation is 9x2+25y2+25z2225=0.

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