JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q20

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MathematicsSingle correct+4 / -1Question 20 of 90

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 $z_1$ is perpendicular to the line through $z_2$ and $z_3$ , then $\arg(z)$ is equal to :

\tan^{-1}\left(\dfrac2{\sqrt5}\right)-\pi

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

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

\tan^{-1}\left(\dfrac34\right)-\pi

Answer and worked solution

Official answer: B

Circle. All three points have modulus 5: C is $|z|=5$ .

Line. $z_2-z_3=4-2i$ , so the perpendicular through $z_1$ has direction $1+2i$ : $z=(3+t)+(4+2t)i$ .

On the circle. $5t^2+22t=0$ , $t=-\frac{22}5$ : $z=-\frac75-\frac{24}5i$ .

Argument (third quadrant): $\tan^{-1}\dfrac{24}7-\pi$ .

(B)

Topics:Geometry in the complex plane Modulus and argument

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JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 20