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  • 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

5.   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

Answers (1)

best_answer

Given,

Two points   A (4, 0, 0) and B (– 4, 0, 0)

let the point P(x,y,z) be a point sum of whose distance from A and B is 10.

So,

The distance PA+The distance PB=10

\sqrt{(x-4)^2+(y-0)^2+(z-0)^2}+\sqrt{(x-(-4))^2+(y)^2+(z)^2}=10

\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}

Squaring on both side :

{(x-4)^2+(y)^2+(z)^2}=100-20\sqrt{(x+4)^2+(y)^2+(z)^2}+{(x+4)^2+(y)^2+(z)^2}

{(x-4)^2-(x+4)^2=100-20\sqrt{(x+4)^2+(y)^2+(z)^2}

-16x=100-20\sqrt{(x+4)^2+(y)^2+(z)^2}

20\sqrt{(x+4)^2+(y)^2+(z)^2}=100+16x

5\sqrt{(x+4)^2+(y)^2+(z)^2}=25+4x

Now again squaring both sides,

25\left ( {(x+4)^2+(y)^2+(z)^2} \right )=625+200x+16x^2

25x^2+200x+400+25y^2+25z^2=625+200x+16x^2

9x^2+25y^2+25z^2-225=0

Hence the equation of the set of points P, the sum of whose distances from A and B is equal to 10 is 9x^2+25y^2+25z^2-225=0.

Posted by

Pankaj Sanodiya

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