Angle between the two spheres $x^{2} + y^{2} + z^{2} + 6 y + 2 z + 8 = 0$ and $x^{2} + y^{2} + z^{2} + 6 x + 8 y + 4 z + 20 = 0$ is

\frac{π}{6}

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\frac{π}{4}

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\frac{π}{3}

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\frac{π}{2}

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Solution

The correct option is A $\frac{π}{2}$
$c_{1} = (0, - 3, - 1), r_{1} = \sqrt{3^{2} + 1^{2} - 8} = \sqrt{2}$
$c_{2} = (- 3, - 4, - 2), r_{2} = \sqrt{3^{2} + 4^{2} + 2^{2} - 20} = 3$
Distance between their centre is $d = \sqrt{3^{2} + 1^{2} + 1^{2}} = \sqrt{11}$
If angle between the given sphere's is $θ$ , then
$cos θ = \frac{r_{1}^{2} + r_{2}^{2} - d^{2}}{2 r_{1} r_{2}} = \frac{2 + 9 - 11}{2 (3) (\sqrt{2})} = 0$
$θ = \frac{π}{2}$

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