If $z$ is a complex number, then $z^{2} + {¯ z}^{2} = 2$ represents-

a circle

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a straight line

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a hyperbola

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an ellipse

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Solution

The correct option is C a hyperbola
Let $z = x + i y$
$¯ z = x - i y$
$∴ z^{2} + {¯ z}^{2} = {(x + i y)}^{2} + {(x - i y)}^{2}$
$= x^{2} + 2 x y i - y^{2} + x^{2} - 2 x y i - y^{2} = 2$
$∴ 2 (x^{2} - y^{2}) = 2$
$∴ x^{2} - y^{2} = 1$
Equation of hyperbola.

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