Question

A point moves so that the sum of its distances from the points (4, 0, 0) and (-4, 0, 0) remains 10. The locus of the point is
[MP PET 1988]

• A. $9x^2+25y^2+25z^2=225$
• B. $9x^2+25y^2-25z^2-225$
• C. $9x^2+25y^2+25z^2=225$
• D. $9x^2+25y^2+25z^2-225$

Answer 1

The correct option is C

$9x^2+25y^2+25z^2=225$

$\sqrt{(x-4{)}^2+y^2+z^2}+\sqrt{(x+4{)}^2+y^2+z^2}=10\Rightarrow 2(x^2+y^2+z^2)+0\sqrt{\left[\right(x-4{)}^2+y^2+z^2]\left[\right(x+4{)}^2+y^2+z^2]}=100-32=68\Rightarrow (x^2+y^2+z^2-34{)}^2=\left[\right(x-4{)}^2+y^2+z^2]\left[\right(x+4{)}^2+y^2+z^2]\Rightarrow \left[\right(x^2+y^2+z^2+16)-8x]\left[\right(x^2+y^2+z^2+16)+8x]=(x^2+y^2+z^2+16{)}^2-4x^2=(x^2+y^2+z^2)+32(x^2+y^2+z^2)-64x^2+(16{)}^2\Rightarrow 9x^2+25y^2+25z^2-225=0.$