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The curve represented by Im (z2) = K where K is a non-zero real number is
- a)a pair of straight lines
- b)an ellipse
- c)a parabola
- d)a hyperbola
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The curve represented by Im (z2) = K where K is a non-zero real number...
Explanation:
General Form of a Complex Number:
- A complex number z can be represented as z = x + iy, where x is the real part and y is the imaginary part.
Given Equation:
- Im(z^2) = K represents the imaginary part of z^2 is equal to a non-zero real number K.
Express z^2 in terms of x and y:
- Let z = x + iy
- z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy
- Im(z^2) = 2xy
Substitute back into the given equation:
- 2xy = K
- xy = K/2
Comparison with Standard Form of Hyperbola:
- The equation xy = constant represents a hyperbola in the xy-plane.
- As the given equation xy = K/2 represents a hyperbola, the curve represented by Im(z^2) = K is a hyperbola.
Therefore, the correct answer is option 'D' - a hyperbola.
General Form of a Complex Number:
- A complex number z can be represented as z = x + iy, where x is the real part and y is the imaginary part.
Given Equation:
- Im(z^2) = K represents the imaginary part of z^2 is equal to a non-zero real number K.
Express z^2 in terms of x and y:
- Let z = x + iy
- z^2 = (x + iy)^2 = x^2 - y^2 + 2ixy
- Im(z^2) = 2xy
Substitute back into the given equation:
- 2xy = K
- xy = K/2
Comparison with Standard Form of Hyperbola:
- The equation xy = constant represents a hyperbola in the xy-plane.
- As the given equation xy = K/2 represents a hyperbola, the curve represented by Im(z^2) = K is a hyperbola.
Therefore, the correct answer is option 'D' - a hyperbola.
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The curve represented by Im (z2) = K where K is a non-zero real number isa)a pair of straight linesb)an ellipsec)a parabolad)a hyperbolaCorrect answer is option 'D'. Can you explain this answer?
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