Let's approach the problem step-by-step:
### Part (a)
We are given two fixed points, A(2, 0) and B(-2, 0). We need to find the locus of point P(x, y) such that the sum of the squares of the distances from P to A and B is 16.
1. Define the distances:
- The distance from P(x, y) to A(2, 0) is given by:
[tex]\[
\sqrt{(x - 2)^2 + y^2}
\][/tex]
- The distance from P(x, y) to B(-2, 0) is given by:
[tex]\[
\sqrt{(x + 2)^2 + y^2}
\][/tex]
2. Sum of the squares of these distances:
[tex]\[
\left(\sqrt{(x - 2)^2 + y^2}\right)^2 + \left(\sqrt{(x + 2)^2 + y^2}\right)^2 = 16
\][/tex]
3. Simplify the expression:
[tex]\[
(x - 2)^2 + y^2 + (x + 2)^2 + y^2 = 16
\][/tex]
4. Expand the squares and combine like terms:
[tex]\[
(x^2 - 4x + 4) + y^2 + (x^2 + 4x + 4) + y^2 = 16
\][/tex]
[tex]\[
x^2 - 4x + 4 + y^2 + x^2 + 4x + 4 + y^2 = 16
\][/tex]
5. Combine like terms:
[tex]\[
2x^2 + 2y^2 + 8 = 16
\][/tex]
6. Simplify:
[tex]\[
2x^2 + 2y^2 = 8
\][/tex]
[tex]\[
x^2 + y^2 = 4
\][/tex]
So, the locus of the point P is a circle with equation:
[tex]\[
x^2 + y^2 = 4
\][/tex]
This is a circle with center at the origin (0, 0) and radius 2.
### Part (b)
The information required to complete part (b) fully is not provided in the problem statement. However, if we assume the problem intends to explore a scenario for the moving point P under a different condition, without the complete specification, we can't determine the exact locus.
Given the known conditions from part (a), we have the required details:
For part (a):
The locus of point 'P' is a circle with equation [tex]\[
x^2 + y^2 = 4\][/tex] and radius 2 centered at the origin (0, 0).
