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For the real parameter t, the locus of the complex number z = (1 – t²) + i√(1 + t2) in the complex plane is
- a)an ellipse
- b)a parabola
- c)a circle
- d)a hyperbola
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
For the real parameter t, the locus of the complex number z = (1 &ndas...
Let z = x + iy
x = 1 – t2
y2 = 1 + t2
Thus, x + y2 = 2
y2 = 2 – x
y2 = – (x – 2)
Thus parabola
Most Upvoted Answer
For the real parameter t, the locus of the complex number z = (1 &ndas...
Explanation:
Complex Number Representation:
The complex number z = (1 – t^2) + i√(1 + t^2) can be written in the form z = x + iy, where x = 1 – t^2 and y = √(1 + t^2).
Parametric Equations:
We can rewrite the above representation in terms of parametric equations:
x = 1 – t^2
y = √(1 + t^2)
Eliminating the Parameter:
By squaring both sides of the equation x = 1 – t^2, we get t^2 = 1 – x. Substituting this into the equation y = √(1 + t^2), we get y = √(2 - x).
Graphical Representation:
The parametric equations x = 1 – t^2 and y = √(1 + t^2) represent a parabolic curve in the complex plane. The locus of the complex number z = (1 – t^2) + i√(1 + t^2) is a parabola.
Therefore, the correct answer is option b) a parabola.
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For the real parameter t, the locus of the complex number z = (1 – t²) + i√(1 + t2)in the complex plane isa)an ellipseb)a parabolac)a circled)a hyperbolaCorrect answer is option 'B'. Can you explain this answer?
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