Q213 / 270 · NDA 2022 (Phase 2) · Mathematics

Consider the equation of a sphere $x^2+y^2+z^2-4 x-6 y-$ $8 z-16=0$. Which of the following statements is/are correct? 1. $z$-axis is tangent to the sphere. 2. The centre of the sphere lies on the plane $x+y+z-9$ $=0$. Select the correct answer using the code given below :

  • A
    1 only
  • Correct answer
    B
    2 only
  • C
    Both 1 and 2
  • D
    Nither 1 nor 2.

Solution

$\begin{aligned} & x^2+y^2+z^2-4 x-6 y-8 z-16=0 \\ & \Rightarrow\left(x^2-4 x+4\right)+\left(y^2-6 y+9\right)+\left(z^2-8 z+16\right) \\ & =16+4+9+16 \\ & \Rightarrow \quad(x-2)^2+(y-3)^2+(z-4)^2=(3 \sqrt{5})^2 \\ & \therefore \quad \text { Centre of sphere }=(2,3,4) \\ & \text { Radius }=3 \sqrt{5} \end{aligned}$ Since, radius $3 \sqrt{5}$ is not equal to $\sqrt{2^2+3^2}=\sqrt{13}$ $\therefore \quad Z$-axis is not tangent to the sphere, so, statement 1 is wrong. $\because 2+3+4-9=0$ $\therefore \quad$ Centre of the sphere lies on the plane $x+y+z-9$ $=0$ So, statement 2 is correct.

Hence, the correct answer is option: 'B'.

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