Toronto Math Forum
MAT3342020S => MAT334Tests and Quizzes => Quiz 1 => Topic started by: Ziyi Wang on January 29, 2020, 11:07:52 AM

Questions:
Write the equation of the perpendicular bisector of the line segment joining $1+2i$ and $12i$.
Answer:
Let $z = x + iy$.
Since $z$ is an arbitrary point on the perpendicular bisector of the line segment joining $1+2i$ and $12i$, we know that the distance between $z$ and $1+2i$ is same as the distance between $z$ and $12i$.
Then, we can calculate that:
$$z(1+2i) = z  (12i)$$
$$(x + iy)(1+2i) = (x + iy)  (12i)$$
$$(x + 1)+(y2)i = (x  1) + (y + 2)i$$
$$\sqrt{(x+1)^2 + (y2)^2} = \sqrt{(x  1)^2 + (y + 2)^2}$$
$$(x+1)^2 + (y2)^2 = (x  1)^2 + (y + 2)^2$$
$$x^2 + 2x + 1 + y^2  4y + 4 = x^2  2x + 1 + y^2 + 4y + 4$$
$$4x = 8y$$
$$y = \frac{1}{2} x$$
Then, we write the equation in complex number notation:
$$Re((\frac{1}{2} + i)z) = 0$$