"Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then $arg(z)$ is equal to :"

Question

Accepted Answer

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${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$

Clearly, $C \equiv {x^2} + {y^2} = 25$

Let $z(x,y)$

$$ \Rightarrow \left( {{{y - 4} \over {x - 3}}} \right)\left( {{2 \over { - 4}}} \right) = - 1$$

$\Rightarrow y = 2x - 2 \equiv L$

$\therefore$ z is intersection of C & L

$\Rightarrow z \equiv \left( {{{ - 7} \over 5},{{ - 24} \over 5}} \right)$

$\therefore$ $Arg(z) = - \pi + {\tan ^{ - 1}}\left( {{{24} \over 7}} \right)$

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Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :

Solution

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Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z1 is perpendicular to the line through z2 and z3, then $arg(z)$ is equal to :

Solution

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More Complex Numbers and Quadratic Equations questions

Drill 25 more like these. Every day. Free.

Let a circle C in complex plane pass through the points ${z_1} = 3 + 4i$, ${z_2} = 4 + 3i$ and ${z_3} = 5i$. If $z( \ne {z_1})$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $arg(z)$ is equal to :