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Let a circle C in complex plane pass through the points z_{1}=3+4 i, z_{2}=4+3 i \text { and } z_{3}=5 i.If z\left(\neq z_{1}\right) is a point on C such that the line through z and z_{1} is perpendicular to the line through z_{2} and z_{3}, then arg \mathrm{\left ( z \right )} is equal to:

Option: 1

\tan ^{-1}\left(\frac{2}{\sqrt{5}}\right)-\pi


Option: 2

\tan ^{-1}\left(\frac{24}{7}\right)-\pi


Option: 3

\tan ^{-1}(3)-\pi


Option: 4

\tan ^{-1}\left(\frac{3}{4}\right)-\pi


Answers (1)

best_answer


\mathrm{\text{Slope of BC }= \frac{3-5}{4-0}= \frac{-1}{2}}
Slope of AP \mathrm{= 2}

\mathrm{eqn\, of\, AP:y-4= 2\left ( x-3 \right )}
\mathrm{\Rightarrow y= 2\left ( x-1 \right )}
P lies on circle  \mathrm{x^{2}+y^{2}= 25}
\mathrm{\Rightarrow x= \frac{-7}{5}\: and\; y= \frac{-24}{5}}
\mathrm{\Rightarrow arg\left ( z \right )= \tan^{-1}\left ( \frac{24}{7} \right )-\pi}

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