Question

find the equation of the set of point P whose sum of distance from A(4,0,0) and B(-4,0,0) is 10

Answer 1

hey friend this is the answer
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Answer 2

The locus of a point P(x,y,z) that has the sum of distances from two points A(4,0,0) and B(-4,0,0) being 10, is an ellipsoid in 3-d and an ellipse in 2-d.

$\sqrt{(x-4)^2+y^2+z^2}+\sqrt{(x+4)^2+y^2+z^2}=10\\\\\sqrt{(x-4)^2+y^2+z^2}=10-\sqrt{(x+4)^2+y^2+z^2}\\\\square \: and \: expand\\\\x^2-8x+16+y^2+z^2 =100+x^2+8x+16+y^2+z^2\\-20 \sqrt{x^2+8x+16+y^2+z^2}\\\\4x+25=5 \sqrt{x^2+8x+16+y^2+z^2}\\\\16x^2+625+200x=25(x^2+8x+16+y^2+z^2)\\\\9x^2+25y^2+25z^2=225\\\\\frac{x^2}{25}+\frac{y^2}{9}+\frac{z^2}{9}=1$

Its equation is:
   x²/25 + y²/9 + z²/9 = 1

It has a major axis of 5*2 and minor axis of 3*2.