$26 \cdot 9$ Some points $P$ are within the region enclosed by a circle of radius 1, and the sum of the squares of the distances from these $P$ points to the endpoints of a certain diameter is 3. Let $n$ be the number of $P$ points, then $n$ is
(A) 0.
(B) 1.
(C) 2.
(D) 4.
(E) Infinite.
(20th American High School Mathematics Examination, 1969)
[Solution] In the Cartesian coordinate system, the equation of a circle with radius 1 and center at the origin is
$$
x^{2}+y^{2}=1 \text {. }
$$

Let $A B$ be the diameter, then the coordinates of $A, B$ are
$$
(-1,0),(1,0) \text {. }
$$

Suppose the coordinates of point $P$ are $(x, y)$. According to the problem, we have
$$
\left[(x+1)^{2}+y^{2}\right]+\left[(x-1)^{2}+y^{2}\right]=3 \text {, }
$$

which simplifies to $x^{2}+y^{2}=\frac{1}{2}$. Therefore, all points on this circle are the solutions.
Hence, the answer is $(E)$.