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NDA2026MathematicsThree Dimensional GeometryActual

Passage: The equation of the sphere S is x^2+y^2+z^2-4x-6y-12z+k=0 . Question: If the radius of the sphere S is 8 units, what is the value of k ?

Options

  1. A-15
  2. B7
  3. C10
  4. D15

Correct answer

A. -15

Step-by-step solution

The given equation of the sphere is x^2+y^2+z^2-4x-6y-12z+k=0 . Comparing this with the general equation of a sphere x^2+y^2+z^2+2ux+2vy+2wz+d=0 , we get: u = -2 , v = -3 , w = -6 , and d = k . The radius of the sphere is given by r = u^2+v^2+w^2-d . Substituting the given values: 8 = (-2)^2+(-3)^2+(-6)^2-k 8 = 4+9+36-k 8 = 49-k Squaring both sides: 64 = 49-k k = 49-64 = -15 Answer: -15

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