Correct Answer - Option 3 : Circle
Formula used:
z = a + ib
\(\rm \overline{z} = a - ib\)
Equation of circle: (x - h)2 + (y - k)2 = r2
Where (h,k) is the coordinates of center of the circle and r is the radius.
Calculation:
we have \(z \overline{z} + |z|^2 + 4(z + \overline{z})-48 = 0\)
(x + iy)(x - iy) + x2 + y2 + 4(x + iy + x - iy) - 48 = 0
⇒ x2 + y2 + x2 + y2 + 4 × 2x - 48 = 0
⇒ 2(x2 + y2) + 8x - 48 = 0
⇒ x2 + y2 + 4x - 24 = 0
⇒ (x + 2)2 + (y - 0)2 = \(\rm (\sqrt{28})^{2}\)
Since, the above equation is similar to (x - h)2 + (y - k)2 = r2
∴ The given equation represent a circle.