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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Three Dimensional Geometry

NDA 2022 · Mathematics · Three Dimensional Geometry

Consider the equation of a sphere x^2+y^2+z^2-4 x-6 y-8 z-16=0 ....

Consider the equation of a sphere x2+y2+z2−4x−6y−8z−16=0x^2+y^2+z^2-4 x-6 y-8 z-16=0. Which of the following statements is/are correct? 1. zz-axis is tangent to the sphere. 2. The centre of the sphere lies on the plane x+y+z−9=0x+y+z-9=0. Select the correct answer using the code given below :

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNither 1 nor 2.

Correct answer

B. 2 only

Step-by-step solution

x2+y2+z2−4x−6y−8z−16=0⇒(x2−4x+4)+(y2−6y+9)+(z2−8z+16)=16+4+9+16⇒(x−2)2+(y−3)2+(z−4)2=(35)2∴ Centre of  sphere =(2,3,4) Radius =35\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\text{ of } sphere }=(2,3,4) \\ & \text{ Radius }=3 \sqrt{5} \end{aligned} Since, radius 353 \sqrt{5} is not equal to 22+32=13∴Z\sqrt{2^2+3^2}=\sqrt{13}\therefore \quad Z-axis is not tangent to the sphere, so, statement 1 is wrong. ∵2+3+4−9=0∴\because 2+3+4-9=0\therefore \quad Centre of the sphere lies on the plane x+y+z−9=0x+y+z-9=0 So, statement 2 is correct.
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