Complex Number

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #61

In complex plane, three points are represented by three complex numbers `z_1, z_2, z_3` and `|z_1| = |z_2| = |z_3| = 1` and `z_1 + z_2 + z_3 = 0`. Then find the area of triangle formed by these three points.

Question No: #62

In a complex plane three points, represented by three complex numbers `z_1, z_2, z_3` and `\frac{1}{z_1-z_2} + \frac{1}{z_2-z_3} + \frac{1}{z_3-z_1} = 0`. Show that the triangle formed by these three points are equilateral triangle.

Question No: #63

In a complex plane, the three points `z_1, z_2, z_3` are three vertices of an isosceles right angle triangle with right angle at `z_3`. Then show that `z_1^2 + 2z_2^2 + z_3^2 = 2z_2(z_1 + z_3)`

Question No: #64

`z_1, z_2` are two complex numbers and `z_1^2 + z_2^2 + 2z_1z_2cos\theta=0`. Then show that the triangle formed by origin, `z_1` and `z_2` is an isosceles triangle. (where `\theta \in \mathbb{R}`)

Question No: #65

If `z = x+iy`, `w = \frac{1-iz}{z-i}` and `|w|=1`, then show that `z` lies on real axis on the complex plane.

Question No: #66

Show that the three points `1+4i, 2+7i` and `3+10i` are collinear.

Question No: #67

If three points `z, -iz` and 1 are collinear, then show that `z` always lies on a circle.

Question No: #68

If `z = x+iy` and `\text{arg}(\frac{z-1}{z+1})=\pi/2`, then show that locus of `z` in complex plane is a circle.

Question No: #69

If `z = x+iy` and `\text{arg}(\frac{z-1}{z+1})=\pi/4`, then show that locus of `z` in a complex plane is a circle.

Question No: #70

If `z_1 = 4+3i`, `z_2 = 7+4i` and `z` is another complex number such that `\text{arg}(\frac{z_1-z}{z-z_2}) = \pi/4`; then show that `|z-6-2i| = \sqrt{5}`