Answer
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Hint: If z is a complex number then \[\bar z\] is represented as its conjugate. Take any general representation of a complex number and find its conjugate then put it in the equation given to solve it to the end. You will get a final equation.
Complete Step by Step Solution:
Let us represent our complex number z as \[x + iy\] clearly the conjugate \[\bar z\] will be \[x - iy\]. Now for solving this put all the values in the equation given. Thus, the equation becomes,
\[\begin{array}{l}
{z^2} + {{\bar z}^2} = 2\\
\Rightarrow {(x + iy)^2} + {(x - iy)^2} = 2\\
\Rightarrow {x^2} + 2ixy + {(iy)^2} + {x^2} - 2ixy + {(iy)^2} = 2\\
\Rightarrow 2{x^2} + 2{i^2}{y^2} = 2\\
Now,i = \sqrt { - 1} \\
{i^2} = - 1\\
\therefore 2{x^2} + 2 \times ( - 1){y^2} = 2\\
\Rightarrow {x^2} - {y^2} = 1
\end{array}\]
Now the general equation of an hyperbola is given by \[\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1\] clearly a and b are 1 here respectively
Therefore option C is correct.
Note: We have used \[{(a + b)^2} = {a^2} + 2ab + {b^2}\] to solve \[{(x + iy)^2}\& {(x - iy)^2}\] . You can also do this question by taking \[z = r(\cos \theta + i\sin \theta )\] So as you are assuming the complex number in polar form. So you will also get the final equation in polar form so remember to convert it according to your convenience.
Complete Step by Step Solution:
Let us represent our complex number z as \[x + iy\] clearly the conjugate \[\bar z\] will be \[x - iy\]. Now for solving this put all the values in the equation given. Thus, the equation becomes,
\[\begin{array}{l}
{z^2} + {{\bar z}^2} = 2\\
\Rightarrow {(x + iy)^2} + {(x - iy)^2} = 2\\
\Rightarrow {x^2} + 2ixy + {(iy)^2} + {x^2} - 2ixy + {(iy)^2} = 2\\
\Rightarrow 2{x^2} + 2{i^2}{y^2} = 2\\
Now,i = \sqrt { - 1} \\
{i^2} = - 1\\
\therefore 2{x^2} + 2 \times ( - 1){y^2} = 2\\
\Rightarrow {x^2} - {y^2} = 1
\end{array}\]
Now the general equation of an hyperbola is given by \[\dfrac{{{x^2}}}{{{a^2}}} - \dfrac{{{y^2}}}{{{b^2}}} = 1\] clearly a and b are 1 here respectively
Therefore option C is correct.
Note: We have used \[{(a + b)^2} = {a^2} + 2ab + {b^2}\] to solve \[{(x + iy)^2}\& {(x - iy)^2}\] . You can also do this question by taking \[z = r(\cos \theta + i\sin \theta )\] So as you are assuming the complex number in polar form. So you will also get the final equation in polar form so remember to convert it according to your convenience.
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