Let's isolate \( y^2 \) in the given equation:
[tex]\[ (x-2)^2 + y^2 = 64 \][/tex]
First, we subtract \( (x-2)^2 \) from both sides of the equation:
[tex]\[ y^2 = 64 - (x-2)^2 \][/tex]
Next, we expand \( (x-2)^2 \):
[tex]\[ (x-2)^2 = x^2 - 4x + 4 \][/tex]
Now we substitute the expanded form back into the equation:
[tex]\[ y^2 = 64 - (x^2 - 4x + 4) \][/tex]
Simplify by distributing the negative sign:
[tex]\[ y^2 = 64 - x^2 + 4x - 4 \][/tex]
Combine like terms:
[tex]\[ y^2 = 60 - x^2 + 4x \][/tex]
So, we have:
[tex]\[ y^2 = -x^2 + 4x + 60 \][/tex]
Therefore, the correct answer matches option B:
[tex]\[ y^2 = -x^2 + 4x + 60 \][/tex]
