Question

Geometrically Im (z² – i) = 2, where \(\rm i = \sqrt { - 1} \) and Im is the imaginary part, represents
1. Circle
2. Ellipse
3. Hyperbola
4. Rectangular hyperbola

Answer 1

Correct Answer - Option 4 : Rectangular hyperbola

CONCEPT:

If, z = x + iy be a complex number, Where x is called the real part of the complex number or Re (z) and y is called the Imaginary part of the complex number or Im (z)

Rectangular Hyperbola: xy = c2

 

CACLCULATION:

The given equation is,

Im(Z² - i) = 2                 ...eq.(1)

Let's simplify the L.H.S first,

Let, Z = x + iy

Then, Z² = ( x + iy)²

  = x² + (iy)² + 2xyi

 = x² - y² + 2xyi

⇒ Z² - i = x2 - y2 + (2xy - 1)i                   ...eq.(2)

Im(Z2 - i) = Imaginary part of (Z2 - i) = (2xy - 1)               ...from eq.(2) 

So, from the above equation we get,

Im(Z2 - i) = (2xy - 1)

Putting this value of Im(Z2 - i)  in eq.(1),

2xy - 1 = 2

⇒ 2xy = 3

⇒ xy = \(\frac{3}{2}\) = \(=( \sqrt{\frac{3}{2}})^2\)

This is a rectangular hyperbola.