If $z$ be complex number such that equation $| z - a^{2} | + | z - 2 a | = 3$ always represents an ellipse then range of value of $a (\in R^{+})$ is

(1, \sqrt{2})

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[1, \sqrt{3}]

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(- 1, 3)

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(0, 3)

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Solution

The correct option is B $(0, 3)$
$I n an ellipse with focus S_{1}, S_{2}, P is any point of ellipse then$
${PS}_{1} + P S_{2} = 2 a$
$L e t S_{1} = a^{2}, S_{2} = 2 a$
$t h e n$
$∣ ∣ a^{2} - 2 a ∣ ∣ < 3$
$a^{2} - 2 a < 3, 2 a - a^{2} < 3$
$\Rightarrow (a - 3) (a + 1) < 0, a^{2} - 2 a + 3 > 0$
$\Rightarrow a \in (- 1, 3), a \in R$
$g i v e n a \in R$
$a \in (0, 3)$

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