Spheres $x^{2} + y^{2} + z^{2} + x + y + z - 1 = 0$ and $x^{2} + y^{2} + z^{2} + x + y + z - 5 = 0$

intersect in a plane

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intersect in five points

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intersect in ten points

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do not intersect

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Solution

The correct option is D do not intersect
The two given equations differ by a constant only.
Hence, the two given spheres have the same center with different radius.
Hence, the they do not intersect.
We can also check it by looking on the equations.
Let $(a, b, c)$ belongs to the intersection of two spheres.
Then, we have $a^{2} + b^{2} + c^{2} + a + b + c - 1 = 0 = a^{2} + b^{2} + c^{2} + a + b + c - 5$ .
$\Rightarrow 4 = 0$ .
An obvious contradiction.

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