Question

Assertion (A) : If the spheres $x^{2}+y^{2}+z^{2}=64$ & $x^{2}+y^{2}+z^{2}-12x+4y-6z+48=0$ touch externally
Reason (R) : If the distance between the centres of spheres is equal to the sum of their radii, then they touch externally

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A)
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (D)

$\because C_{1} \equiv (0, 0, 0), r_{1} = 8$
$C_{2} \equiv\left(6, -2, 3\right), r_{2}=\sqrt{36+4+9-48}=1$
As $C_{1}C_{2}=\sqrt{36+4+9}=7$ & $\left|r_{1}-r_{2}\right|=7$
$\Rightarrow C_{1}\,C_{2} =\left|r_{1}-r_{2}\right|=7$
$\Rightarrow $ Spheres touches internally
$\Rightarrow $ Assertion (A) is false but Reason (R) is true