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Geometrically Re(z2 − i) = 2 , where i = √−1 and Re is the real part, represents
- a)Circle
- b)Ellipse
- c)Rectangular hyperbola
- d)Parabola
Correct answer is option 'C'. Can you explain this answer?
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Geometrically Re(z2 − i) = 2 , where i = √−1 and Re ...
This equation represents rectangular hyperbola.
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Geometrically Re(z2 − i) = 2 , where i = √−1 and Re ...
Rectangular Hyperbola Explanation
The equation Re(z^2 - i) = 2 represents a rectangular hyperbola. Let's break down the equation and understand why it represents a rectangular hyperbola:
Complex Number Representation:
- Let z = x + yi, where x is the real part and yi is the imaginary part.
- Re(z^2 - i) = 2 can be written as Re((x + yi)^2 - i) = 2.
Expand the Equation:
- (x + yi)^2 = x^2 + 2xyi - y^2
- Re(x^2 + 2xyi - y^2 - i) = 2
- x^2 - y^2 = 2
Standard Form of Hyperbola:
- The equation x^2 - y^2 = 2 is in the standard form of a rectangular hyperbola.
- In general, the equation of a rectangular hyperbola is x^2/a^2 - y^2/b^2 = 1, where a and b are constants.
Conclusion:
- The equation Re(z^2 - i) = 2 represents a rectangular hyperbola with the center at the origin and the x-axis as the transverse axis.
- Hence, the correct answer is option 'C' - Rectangular hyperbola.
The equation Re(z^2 - i) = 2 represents a rectangular hyperbola. Let's break down the equation and understand why it represents a rectangular hyperbola:
Complex Number Representation:
- Let z = x + yi, where x is the real part and yi is the imaginary part.
- Re(z^2 - i) = 2 can be written as Re((x + yi)^2 - i) = 2.
Expand the Equation:
- (x + yi)^2 = x^2 + 2xyi - y^2
- Re(x^2 + 2xyi - y^2 - i) = 2
- x^2 - y^2 = 2
Standard Form of Hyperbola:
- The equation x^2 - y^2 = 2 is in the standard form of a rectangular hyperbola.
- In general, the equation of a rectangular hyperbola is x^2/a^2 - y^2/b^2 = 1, where a and b are constants.
Conclusion:
- The equation Re(z^2 - i) = 2 represents a rectangular hyperbola with the center at the origin and the x-axis as the transverse axis.
- Hence, the correct answer is option 'C' - Rectangular hyperbola.
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Geometrically Re(z2 − i) = 2 , where i = √−1 and Re is the real part, representsa)Circleb)Ellipsec)Rectangular hyperbolad)ParabolaCorrect answer is option 'C'. Can you explain this answer?
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