To determine the equation of the locus of a moving point whose sum of the squares of the distances from two fixed points is a constant, let's analyze the given condition step-by-step:
1. Define the points and variables:
Let the fixed points be \( (a, 0) \) and \( (-a, 0) \). Let \( (x, y) \) be the coordinates of the moving point.
2. Express the distances:
The distance from the moving point \( (x, y) \) to \( (a, 0) \) is given by:
[tex]\[
\sqrt{(x - a)^2 + y^2}
\][/tex]
Similarly, the distance from the moving point \( (x, y) \) to \( (-a, 0) \) is:
[tex]\[
\sqrt{(x + a)^2 + y^2}
\][/tex]
3. Apply the given condition:
According to the problem, the sum of the squares of these distances is \( 2c^2 \):
[tex]\[
\left( \sqrt{(x - a)^2 + y^2} \right)^2 + \left( \sqrt{(x + a)^2 + y^2} \right)^2 = 2c^2
\][/tex]
4. Simplify the equation:
[tex]\[
(x - a)^2 + y^2 + (x + a)^2 + y^2 = 2c^2
\][/tex]
5. Expand and combine like terms:
[tex]\[
x^2 - 2ax + a^2 + y^2 + x^2 + 2ax + a^2 + y^2 = 2c^2
\][/tex]
6. Combine the terms:
[tex]\[
x^2 + x^2 - 2ax + 2ax + a^2 + a^2 + y^2 + y^2 = 2c^2
\][/tex]
[tex]\[
2x^2 + 2a^2 + 2y^2 = 2c^2
\][/tex]
7. Factor out the common term and divide by 2:
[tex]\[
x^2 + a^2 + y^2 = c^2
\][/tex]
8. Rearrange the equation:
[tex]\[
x^2 + y^2 = c^2 - a^2
\][/tex]
Therefore, the correct equation for the locus of the moving point under the given conditions is:
[tex]\[ x^2 + y^2 = c^2 - a^2 \][/tex]
Thus, the correct answer is:
2) [tex]\( x^2 + y^2 = c^2 - a^2 \)[/tex]
