If z_1 = 10 + 6i,z_2 = 4 + 6i and z is any co | vedclass

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Question

(A) Given $	ext{arg}\left(\frac{z - z_1}{z - z_2}ight) = \frac{\pi}{4}$.This represents the locus of $z$ as an arc of a circle passing through $z_1

Vedclass · Accepted Answer

$|z - 7 - 9i| = 3\sqrt{2}$

Vedclass · Answer

(A) Given $	ext{arg}\left(\frac{z - z_1}{z - z_2}ight) = \frac{\pi}{4}$.This represents the locus of $z$ as an arc of a circle passing through $z_1$ and $z_2$.The angle subtended by the chord $z_1z_2$ at the circumference is $\frac{\pi}{4}$,so the angle subtended at the center $O$ is $2 	imes \frac{\pi}{4} = \frac{\pi}{2}$.The midpoint of $z_1z_2$ is $\left(\frac{10+4}{2}, \frac{6+6}{2}ight) = (7, 6)$.The distance between $z_1$ and $z_2$ is $|10+6i - (4+6i)| = 6$. Thus,half the chord length is $3$.Since the triangle formed by the center and the chord is an isosceles right triangle,the distance from the midpoint $(7, 6)$ to the center $O$ is $3$.Since the angle is $\frac{\pi}{4}$,the center $O$ is at $(7, 6+3) = (7, 9)$,which corresponds to $7+9i$.The radius $R$ is the distance from $(7, 9)$ to $(10, 6)$,which is $\sqrt{(10-7)^2 + (6-9)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2}$.Thus,the equation of the circle is $|z - (7+9i)| = 3\sqrt{2}$.

If $z_1 = 10 + 6i$,$z_2 = 4 + 6i$ and $z$ is any complex number such that the argument of $\frac{z - z_1}{z - z_2}$ is $\frac{\pi}{4}$,then

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