The number of common tangent planes to the spheres $(x+2)^2+y^2+z^2=1 , (x-2)^2+y^2+z^2=1$ passing through the origin is equal to

Question

The number of common tangent planes to the spheres $(x+2)^2+y^2+z^2=1 , (x-2)^2+y^2+z^2=1$ passing through the origin is equal to

(A) 0 (B) 1 (C) 2 (D) None of these .

My work :

Drawing the graph it seems the spheres do not touch each other . Hence has no common tangent . This is also follows from the equation : Solving the two equations we get $x=0$ . So if the spheres are touching each other at the point $(o,\alpha , \beta) $ then $(0-2)^2+\alpha^2+\beta^2=1 $ which clearly does not have any real solution .

Hence the answer must be (A) $0$ .

Is my solution correct ?

Answer 1

There are infinitely many lines which are tangent to one of the sphere $S_1$ and pass through the origin. By symmetry, such lines will also be tangent to the another sphere $S_2$.

Let $l$ be one of such line and that it touches $S_1$ at point $p$. Consider the tangent plane $P$ of $S_1$ at $p$. Then $P$ contains $l$ and is tangent to $S_2$ by symmetry.

Hence there should be infinitely many of them and the answer is (D).

Answer 2

The spheres are with centers $C_1(x_1,y_1,z_1)=(2,0,0)$ and $c_2(x_2,y_2,z_2)=(-2,0,0)$ with common radius 2.

There are two categories of planes that are tangent to these spheres.

First category:

Let us define plane $P_{\theta}$ (passing through the origin) by equation:

$$-x+\sqrt{3}\cos(\theta)y+\sqrt{3}\sin(\theta)z=0$$

$P_{\theta}$ is tangent to the first sphere because distance $d$ from $C_1$ to this plane is equal to 1 (the radius):

$$d=\dfrac{|-x_1+\sqrt{3}\cos(\theta)y_1+\sqrt{3}\sin(\theta)z_1|}{\sqrt{1^2+\sqrt{3}^2 \cos^2(\theta)+\sqrt{3}^2\sin^2(\theta)}}=\dfrac{2}{\sqrt{4}}=1$$

using formula (9) in (http://mathworld.wolfram.com/Point-PlaneDistance.html).

The same for $C_2$.

Thus there is an infinite number of planes that are simultaneously tangent to both spheres. The answer is therefore (D).

2nd category: (of planes mutually tangent to both spheres) I let you find it (hint: think to the cylinder with axis $Ox$ and radius 1).