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If z = x + iy, where i = √-1, then what does the equations \(\rm z \overline{z} + |z|^2 + 4(z + \overline{z})-48 = 0\) represent?
1. Straight line
2. Parabola
3. Circle
4. Pair of straight lines

1 Answer

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Best answer
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.

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