1969 AMC 12 Problem 28

Question

Let

n

be the number of points

P

interior to the region bounded by a circle with radius

1,

such that the sum of the squares of the distances from

P

to the endpoints of a given diameter is

3.

Then

n

is:

0

1

2

4

infinite

Place the circle at the origin with diameter endpoints

(-1,0)

and

(1,0)

Simplify the sum of the two squared distances

Let

P=(x,y)

and take the diameter endpoints as

(-1,0)

and

(1,0).

The condition becomes

\begin{aligned} PA^2+PB^2&=3,\\ 2x^2+2y^2+2&=3. \end{aligned}

Thus

x^2+y^2=\frac{1}{2}.

This is an entire circle of radius

\frac{1}{\sqrt2},

lying inside the given unit circle. It contains infinitely many points.

Therefore, the correct answer is E.

Problem 28 in other years