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NDA 2023 Maths · Mathematics · Three Dimensional Geometry · medium

The sphere with centre (1,2,3) that touches the xy-plane has the equation

Single correct option

The sphere with centre (1,2,3)(1,2,3) that touches the xyxy-plane has the equation
  1. A. x2+y2+z2−2x−4y−6z+5=0x^2+y^2+z^2-2x-4y-6z+5=0
  2. B. x2+y2+z2−2x−4y−6z+14=0x^2+y^2+z^2-2x-4y-6z+14=0
  3. C. x2+y2+z2−2x−4y−6z−5=0x^2+y^2+z^2-2x-4y-6z-5=0
  4. D. x2+y2+z2−2x−4y−6z=0x^2+y^2+z^2-2x-4y-6z=0
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Answer: (A) x2+y2+z2−2x−4y−6z+5=0x^2+y^2+z^2-2x-4y-6z+5=0

The distance of the centre from the xyxy-plane is 33, so the radius is 33. Expanding (x−1)2+(y−2)2+(z−3)2=9(x-1)^2+(y-2)^2+(z-3)^2=9 gives x2+y2+z2−2x−4y−6z+14−9=0x^2+y^2+z^2-2x-4y-6z+14-9=0, i.e. constant term +5+5. The constant 1414 would give a zero-radius sphere.
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