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Equation of Conics in Complex Form

If the maximu...

Question

If the maximum possible principal argument of the complex number $z$ satisfying $| z - 4 | = R e (z)$ is $k$ , then the value of $\frac{π}{k}$ is

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Solution

Let $z = x + i y$
$| z - 4 | = R e (z) \Rightarrow \sqrt{(x - 4)^{2} + y^{2}} = x \Rightarrow x^{2} - 8 x + 16 + y^{2} = x^{2} \Rightarrow y^{2} = 8 (x - 2)$
This represents a parabola whose directrix is $x = 0$

$arg (z)$ is maximum possible when the line drawn from origin is tangent to the parabola.
As $x = 0$ is directrix, so angle between the pair of tangents is $\frac{π}{2} .$

Therefore, the maximum possible value is
$arg (z) = \frac{π}{4} ∴ \frac{π}{k} = 4$

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Equation of Conics in Complex Form

Standard XII Mathematics

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