JEE MAIN - Mathematics (2022 - 25th June Morning Shift - No. 15)
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 :
$${\tan ^{ - 1}}\left( {{2 \over {\sqrt 5 }}} \right) - \pi $$
$${\tan ^{ - 1}}\left( {{{24} \over 7}} \right) - \pi $$
$${\tan ^{ - 1}}\left( 3 \right) - \pi $$
$${\tan ^{ - 1}}\left( {{3 \over 4}} \right) - \pi $$
Explanation
$${z_1} = 3 + 4i$$, $${z_2} = 4 + 3i$$ 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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