Can you help solve this problem?

Q. Prove the locus of the centre of the circle

1/2(x*x+y*y) +xcosa + ysina -4 =0 is x*x+y*y=1

1/2(x²+y²) + xcosa + ysina - 4 = 0

x²+y² + 2xcosa + 2ysina - 8 =0

coordinates of center = (h,k) = (-g,-f)

 g = cosa , f=sina             (from eq of circle)

(h,k) = (-cosa, -sina)

h = -cosa , k = -sina

h² + k² = (-cosa)² + (-sina)²

            =sin²a+cos²a=1

h² + k² = 1

replacing (h,k) by (x,y)

x²+y² = 1           

this is the required equation

approve my ans if u like it

Approved

Let center of the circle be (h, k) .

center of given circle=(-cos a, -sin a)

(-cos a, -sin a) =(h, k) → h = cos a and k = -sin a

cos²a + sin²a = 1 → h²+k²= 1

locus of (h,k) is x²+y²=1

Approved

Thank you sirs.