Question

Consider the following statements :
1. The two spheres intersect each other.
2. The radius of first sphere is less than that of second sphere.
Which of the above statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (C)

Both 1 and 2

phere 1

$x^2+y^2+z^2-4y+3=0$

Rearranging terms, we get

${(x-0)}^2+{(y-2)}^2+{(z-0)}^2=1$

So, the center of the sphere is at $(0,2,0)$ and the radius is $1$.

Sphere 2

$x^2+y^2+z^2+2x+4z-4=0$

$\Rightarrow {(x+1)}^2+{(y-0)}^2+{(z+2)}^2=9$

Center $(-1,0,-2)$

Radius $=3$

So, distance between the two centers is

$d=\sqrt{(0-(-1){)}^2+{(2-0)}^2+{(0-(-2\left)\right)}^2}=\sqrt{9}=3$

Condition for intersection of two sphere is $r_1-r_2<d<r_1+r_2$

This condition is satisfied.

So, the spheres intersect and radius of first sphere$\left(r_1\right)<$ radius of second sphere $\left(r_2\right)$

Hence, both statements are correct.